Title: Math and science / Analysis / Tao / 06. Limits of sequences
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Title: Cauchy sequences of reals
We can define Cauchy sequences of reals in the same way as was done for rationals. For rationals, many ideas of closeness were introduced separately before reaching Cauchy sequences. This card compresses all of the ideas for reals.
Distance and ε-close reals
Let \( a \) and \( b \) be two real numbers. Define their distance \( d(a,b) \) to be  \( d(a,b) := |a - b| \). Let \( \varepsilon > 0 \) be a real. The two reals \( a \) and \( b \) are said to be ε-close if \( d(a,b) \le \varepsilon \).
Sequence of reals
A sequences of reals is denoted in the same way as a sequences of rationals: \( (a_n)_{n=m}^{\infty} \) represents a function \( f \) from the set \( \{y \in \mathbb{Z} : y \ge m \} \) to the set of reals, where \( a_n := f(n) \).
ε-steady sequence of reals
A sequence \( (a_n)_{n=N}^{\infty} \) of reals is said to be ε-steady if \( a_j \) and \( a_k \) are ε-close for all \( j, k \ge N \).
Eventually ε-steady
A sequence \( (a_n)_{n=m}^{\infty} \) of reals is said to be eventually ε-steady if \( (a_n)_{n=N}^{\infty} \) is ε-steady for some integer \( N \ge m \).
Cauchy sequence
A sequence \( (a_n)_{n=m}^{\infty} \) of reals is said to be a Cauchy sequence if it is eventually ε-steady for every real \( \varepsilon > 0 \).
Cauchy sequences of reals → convergence of sequences of reals → uniqueness of convergence → limit, the definition → subsumption of formal limits
Source: Tao, Analysis I
Chapter 6

Title: Convergence of sequences of reals
Q: Let \( L \) be a real number. A sequence of reals \( (a_n)_{n=m}^{\infty} \) is said to converge to \( L \) if for every real \( \varepsilon > 0 \) there exists an integer \( N \ge m \) such that [...] for all \( k \ge N \).
A: \( |a_k - L| \le \varepsilon \)
BackOnlyText: Expanded definition
The definition can also be approached in now familiar eventual ε-close progression.
ε-close
A sequences of reals \( (a_n)_{n=m}^{\infty} \) is said to be ε-close to \( L \) if \( a_k \) is ε-close to \( L \) for all \( k \ge m \).
eventually ε-close
The sequence is said to be eventually ε-close to \( L \) if \( a_k \) is ε-close to \( L \) for all \( k \ge N \) for some integer \( N \ge m \).
convergence
The sequence is said to converge to \( L \) if it is eventually ε-close to \( L \) for every \( \varepsilon > 0 \).
Rationals vs real sequences
Defining convergence of reals is another step towards defining a limit for reals, which will eventually subsume the original formal limit for rationals, \( LIM_{n\rightarrow} a_n \). For the set of reals, \( \mathbb{R} \), a sequence can approach an element of \( \mathbb{R} \). This is in contrast to the case with rationals, \( Q \), where Cauchy sequences would not necessarily approach a number that is an element of \( Q \); instead, the sequence is used to define a new number \( LIM_{n\rightarrow} a_n \).
Convergent sequences and Cauchy sequences of reals
So far, convergent sequences have been defined independently of any notion of Cauchy sequences of reals. So it has not yet be asked as to whether there is any relation between the two. Further cards will show that a sequence of reals is convergent if and only if it is a Cauchy sequence! We still need the idea of convergent sequences, as it allows us to create an identity between a convergent sequence and a real (aka a limit). The definition of Cauchy sequences in comparison only involves relative close-ness within the sequence and not closeness to some other number, thus it cannot be used to create an equality with a real.
Topological perspective of convergence
Source: Tao, Analysis I
Chapter 6
Abbott, Understanding Analysis
Chapter 2
Context: Cauchy sequences of reals → convergence of sequences of reals → uniqueness of convergence → limit, the definition → subsumption of formal limits

Title: Uniqueness of convergence, proposition
Let \( (a_n)_{n=m}^{\infty} \) be a sequence of reals, and let \( L \ne L' \) be two distinct real numbers. Then \( (a_n)_{n=m}^{\infty} \) cannot converge to both \( L \) and \( L' \).
Proof
If \( L \ne L' \), then there exists a positive real \( d = dist(L, L') = | L - L'| \). As \( (a_n)_{n=m}^{\infty} \) converges to \( L \), we can choose a real \( 0 < \varepsilon < \frac{d}{3} \) and there exists an integer \( N_1 > m \) such that \( (a_n)_{n=N_1}^{\infty} \) is ε-close to \( L \). Similarly, for \( L' \) for the same \( \varepsilon \), there. exists an integer \( N_2 > m \) such that \( (a_n)_{n=N_2}^{\infty} \) is ε-close to \( L' \). Thus, there exists an \( M = \max(N_1, N_2) \) (or just \( M = N_1 + N_2 \), if you don't want to use \( \max \)), such that \( a_k \) is ε-close to both \( L \) and \( L' \) for all \( k \ge M \). As we have both \( dist(a_k, L) \le \varepsilon \) and \( dist(a_k, L') \le \varepsilon \), this implies, by the triangle inequality, that \( dist(L, L') \le 2\varepsilon = \frac{2}{3} dist(L, L') \), which is a false statement, as the distance is always positive. Thus, it cannot be true that \( (a_n)_{n=m}^{\infty} \) converges to both \( L \) and \( L' \).
BackOnlyText: Once it is shown that sequences of reals, if they converge to a real, must converge to a unique real, we can give that unique real a name. It will be called a limit of a sequence (there is a separate card for this definition).
Thus, a limit is simply the real for which a sequence of reals converges to.
Cauchy sequences of reals → convergence of sequences of reals → uniqueness of convergence → limit, the definition → subsumption of formal limits
Source: Tao, Analysis I
Chapter 6

Title: Limits of sequences
If a sequence of reals \( (a_n)_{n=m}^{\infty} \) converges to a real \( L \), we say that \( (a_n)_{n=m}^{\infty} \) is convergent and that \( L \) is the limit of the sequence. We write:
\[ L = \lim_{n\to\infty}a_n \]
If a sequence does not converge to a real, then it is said to diverge or be divergent. \( \lim_{n\to\infty}a_n \) is left undefined for a divergent sequence.
BackOnlyText: \( L \) is unique as per the Uniqueness of Convergence proposition. The uniqueness allows us to create the limit notation \( \lim_{n\to\infty} a_n \), dependent only on the sequence \( (a_n)_{n=m}^{\infty} \), and for that notation to be equated to a single real. If fact, the limit notation doesn't depend on the full specification of the sequence \( (a_n)_{n=m}^{\infty} \), as the starting index is irrelevant and neglected. So the \( \lim_{n\to\infty} a_n \) is the limit for any sequences \( (a_n)_{n=N}^{\infty} \) where \( N \) is some integer.
Example: \( \lim_{n\to\infty} \frac{1}{n} = 0 \), proof
\( \lim_{n\to\infty} \frac{1}{n}= 0 \) is true if the sequence \( (a_n)_{n=1}^{\infty} \) converges to 0, where \( a_n = \frac{1}{n} \).
Let \( \varepsilon > 0 \) be an arbitary real. We must show that there exists an integer \( N \) such that \( |a_n - 0| \le \varepsilon \) for all \( n \ge N \).
\( |a_n - 0| = |\frac{1}{n} - 0| = \frac{1}{n} \le \frac{1}{N} \).
\( \frac{1}{N} \le \varepsilon \) is true if \( N \ge \frac{1}{\varepsilon} \). Such an integer \( N \) exists as per the Achimedean principal. Thus \( |a_n - 0| \le \varepsilon \) for all \( n \ge N \). Thus, \( (a_n)_{n=1}^{\infty} \) is eventually \( \varepsilon \) close to 0 for arbitary \( \varepsilon > 0 \). Thus, \( (a_n)_{n=1}^{\infty} \) converges to 0. This allows us to write \( \lim_{n\to\infty} \frac{1}{n}= 0 \).
Cauchy sequences of reals → convergence of sequences of reals → uniqueness of convergence → limit, the definition → subsumption of formal limits
Source: Tao, Analysis I
Chapter 6

Title: Formal limits are genuine limits
Q: Suppose that \( (a_n)_{n=1}^{\infty} \) is a [...] of rational numbers (defined for reals). Then \( (a_n)_{n=1}^{\infty} \) converges to \( LIM_{n\to\infty}a_n \), i.e.
\[ \lim_{n\to\infty}a_n = LIM_{n\to\infty}a_n \]
A: Cauchy sequence
BackOnlyText: The real defined by the Cauchy sequence is equal to the real to which the matching sequence of reals converges to.
Going forward, this equality allows limits to be used inplace of formal limits. Underneath, however, each real is still defined as a formal limit of rationals.
This masking of the foundations was similarly undertaken for rationals, integers and natural numbers. The pair of integers combined with '//' to make a rational, the pair of natural numbers combined with '—' to make an integer, and the increment operation—all were masked by subsequently defined operations.
Source: Tao, Analysis I
Chapter 6
Context: Cauchy sequences of reals → convergence of sequences of reals → uniqueness of convergence→ limit, the definition → subsumption of formal limits

Title: Limit laws
Let \( (a_n)_{n=1}^{\infty} \) and \( (b_n)_{n=1}^{\infty} \) be sequences of reals that converge to the reals \( x \) and \( y \) respectively. i.e.
\[ \lim_{n\to\infty}a_n = x \\
   \lim_{n\to\infty}b_n = y \]
The 8 basic limit laws are:
1. Addition. The sequence \( (a_n + b_n)_{n=m}^{\infty} \) converges to \( x + y \). In other words,
\[ \lim_{n\to\infty}(a_n + b_n) = \lim_{n\to\infty}a_n + \lim_{n\to\infty}b_n \]
2. Multiplication. The sequence \( (a_n b_n)_{n=m}^{\infty} \) converges to \( xy \). In other words,
\[ \lim_{n\to\infty}(a_n b_n) = (\lim_{n\to\infty}a_n)(\lim_{n\to\infty}b_n) \]
3. Constant multiplication. The sequence \( (c a_n)_{n=m}^{\infty} \) converges to \( cx \). In other words,
\[ \lim_{n\to\infty}(ca_n) = c(\lim_{n\to\infty}a_n) \]
4. Subtraction. The sequence \( (a_n - b_n)_{n=m}^{\infty} \) converges to \( x - y \). In other words,
\[ \lim_{n\to\infty}(a_n - b_n) = \lim_{n\to\infty}a_n - \lim_{n\to\infty}b_n \]
5. Reciprocation. Suppose \( y \ne 0 \) and \( b_n \ne 0 \) for all \( n \ge m \). Then the sequence \( (b_n^{-1})_{n=m}^{\infty} \) converges to \( y^{-1}  \). In other words,
\[ \lim_{n\to\infty}(b_n^{-1} ) = (\lim_{n\to\infty}b_n)^{-1} \]
6. Division. Suppose \( y \ne 0 \) and \( b_n \ne 0 \) for all \( n \ge m \). Then the sequence \( (\frac{a_n}{b_n})_{n=m}^{\infty} \) converges to \( \frac{x}{y} \). In other words,
\[ \lim_{n\to\infty} \frac{a_n}{b_n} = \frac{lim_{n\to\infty} a_n}{lim_{n\to\infty}b_n} \]
7. The sequence \( (max(a_n, b_n))_{n=m}^{\infty} \) converges to \( max(x,y) \); in other words,
\[ \lim_{n\to\infty} max(a_n, b_n) = max(\lim_{n\to\infty} a_n, \lim_{n\to\infty} b_n) \] 
8. The sequence \( (min(a_n, b_n))_{n=m}^{\infty} \) converges to \( min(x,y) \); in other words,
\[ \lim_{n\to\infty} min(a_n, b_n) = min(\lim_{n\to\infty} a_n, \lim_{n\to\infty} b_n) \]
Source: Tao, Analysis I
Chapter 6, p131

Title: Suprema and infima of sequences, definition
Q: Carrying over the idea of supremum and infimum of sets of reals to supremum and inimum of sequences of reals.
Let \( (a_n)_{n=m}^{\infty} \) be a sequence of real numbers. Then we define \( \sup(a_n)_{n=m}^{\infty} \) to be the supremum of [...], and \( \inf(a_n)_{n=m}^{\infty} \) to be the infimum of the same [...].
A: the set \( \{a_n : n \ge m\} \); set \( \{a_n : n \ge m\} \)
BackOnlyText: The quantities \( \sup(a_n)_{n=m}^{\infty} \) and \( \inf(a_n)_{n=m}^{\infty} \) are sometimes written as \( \sup_{n \ge m}a_n \) and \( \inf_{n \ge m}a_n \) respectively.
Sequences as functions
When sequences are viewed as functions from \( \mathbb{N} \) to \( \mathbb{R} \), the supremum of the sequence can be viewed as the supremum of the codomain of the function.

Title: Limit points
Let \( (a_n)_{n=m}^{\infty} \) be a sequence of real numbers, let x be a real number, and let \( \epsilon > 0 \) be a real number. We say that \( x \) is ε-adherent to \( (a_n)_{n=m}^{\infty} \) iff there exists an \( n \ge m \) such that \( a_n \) is ε-close to x. We say that \( x \) is continually ε-adherent to \( (a_n)_{n=m}^{\infty} \) if it is ε-adherent to \( (a_n)_{n=N}^{\infty} \) for every \( N \ge m \). We say that \( x \) is a limit point or adherent point of \( (a_n)_{n=m}^{\infty} \) if it is continually ε-adherent to \( (a_n)_{n=m}^{\infty} \) for every \( \epsilon > 0 \).
How are limit points different to limits?
BackOnlyText: The definition is motivated by a sequence such as:
\[ 1.1, -1.01, 1.001, -1.0001, 1.00001,... \]
The sequence does not converge, but half the time it is getting closer to to 1, and half the time it is getting closer to -1. We wish to name and describe properties of these numbers which are not limits of the squence, but have some qualities of being limits.
How are limit points different to limits?
In the case of limits, ε-close-ness requires all elements of a sequence to be within a certain distance of a number. For limit points, ε-adherance only requires a single element of the sequence to be within a certain distance of a number. To continue the comparison, eventually ε-close-ness requires some \( N \ge m \) such that \( (a_n)_{n=N}^{\infty} \) is ε-close to something, whereas continual ε-adherance requires \( (a_n)_{n=N}^{\infty} \) to be ε-adherant for all \( N \ge m \). So in the first case we have a  \( \exists x\, \forall y\,  (y \ge x) \supset close(target, y) \) predicate and in the latter case we have a  \( \forall x\, \exists y\, (y \ge x) \supset close(target, y) \).
To add to the comparison, it is case that limits are limit points. And if a limit exists, then it is the only limit point. (Proposition 6.4.5)
Example: The above picture uses the verb 'adherant' the wrong way around. Correct: all elements of D' are ε-adherant to the sequence.

Title: Limit superior and limit inferior
Q: Suppose that \( (a_n)_{n=m}^{\infty} \) is a sequence. We define a new sequence \( (a_N^+)_{N=m}^{\infty} \) by the formula:
\[ a_N^+ := \sup(a_n)_{n=N}^{\infty} \]
In other words, \( a_N^+ \) is the supremum of all the elements in the sequence from \( a_N \) onwards.
We then define the limit superior of the sequence \( (a_n)_{n=m}^{\infty} \), denoted by \( \limsup_{n\rightarrow \infty}a_n \), by the formula:
[...]
The mirror proceedure defines the limit inferior. First define:
\[ a_N^- := \inf(a_n)_{n=N}^{\infty} \]
The the limit inferior is:
[...]
A: \[\limsup_{n\rightarrow \infty}a_n := \inf(a_N^+)_{N=m}^{\infty} \]; \[ \liminf_{n\rightarrow \infty}a_n := \sup(a_N^-)_{N=m}^{\infty} \]
BackOnlyText: The above formulations present limit superior as the smallest supremum of all sequences with different starting points. Ditto limit inferior. Limit superior and inferior can be defined instead as limits (which makes their name make more sense):
\[ \limsup_{n\rightarrow \infty} a_n := \lim_{n\rightarrow \infty}  ( \sup_{m\gt n} x_m )\]
Limit superior/inferior, using Tao's piston analogy, are the points where pistons pressing down (superior) and up(inferior) on a sequence will come to rest as the pistons are moved towards infinity along the tail of the sequence.
Example: A sequence with a cosine shape is an example that demonstrates an clear difference between limit points and limit inferior/superior. A constant magnitude cosine will have every value from [-1,1] as a limit point. 1 will be the limit superior, and -1 will be the limit inferior.
Another example:

Title: Supremum of sets of extended reals
We wish to transitioning from sets of reals to sets of extended reals. We define the supremum and infimum of sets of extended reals like so:
Let E be a subset of \( \mathbb{R}^* \). Then we define the supremum of \( sup(E) \) or least upper bound of \( E \) by the following rule.
1. If \( E \) is contained in \( \mathbb{R} \) (i.e., \( +\infty \) and \( -\infty \) are not elements of E), then we let sup(E) be the supremum already defined for sets of reals.
2. If \( E \) contains \( +\infty \), then we set \( sup(E) \) like so: \( sup(E) := +\infty \).
3. If E does not contain \( +\infty \) but does contain \( -\infty \), then we set \( sup(E) := sup(E\setminus \{-\infty\}) \) (which is a subset of \( \mathbb{R} \) and thus falls under case 1).
We also (some what lazily, or craftily) define the infimum \( inf(E) \) of \( E \) (also known as the greatest lower bound of \( E \) by the formula:
\[ inf(E) := -sup(-E) \] 
Where \( -E \) is the set \( -E := \{-x : x \in E\} \).

Title: Squeeze test
Q: Let \( (a_n)_{n=m}^{\infty} \), \( (b_n)_{n=m}^{\infty} \) and \( (c_n)_{n=m}^{\infty} \) be sequences of real numbers such that:
\[ a_n \le b_n \le c_n \]
for all \( n \ge m \). Suppose that \( (a_n)_{n=m}^{\infty} \) and \( (c_n)_{n=m}^{\infty} \) both converge to the same limit \( L \). Then [...].
A: \( (b_n)_{n=m}^{\infty} \) also converges to \( L \)
BackOnlyText: This is not an iff relationship.
This proof is broken down using the comparison principle, then using the fact that if a sequence converges, then its limit superior and limit inferior both equal the limit of the sequence.
The comparison principle
Suppose that \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) are two sequences of real numbers such that \( a_n \le b_n \) for all \( n \ge m \). Then we have the inequalities:
\[
\begin{aligned}
\sup(a_n)_{n=m}^{\infty} &\le \sup(b_n)_{n=m}^{\infty} \\
\inf(a_n)_{n=m}^{\infty} &\le \inf(b_n)_{n=m}^{\infty} \\
\limsup_{n\rightarrow \infty} a_n &\le \limsup_{n\rightarrow \infty} b_n \\
\liminf_{n\rightarrow \infty} a_n &\le \liminf_{n\rightarrow \infty} b_n
\end{aligned}
\]
Source: Tao, Analysis I
Chapter 6, p145

Title: Zero test for sequences
Q: A consequence of the squeeze test is the following:
Let \( (a_n)_{n=m}^{\infty} \). Then the limit \( \lim_{n\rightarrow \infty} a_n \) exists and is equal to zero if and only if the limit [...] exists and is equal to zero.
A: \( \lim_{n \rightarrow \infty} |a_n| \)
BackOnlyText: Proof?

Title: Completeness of the reals, theorem
Q: A sequence \( (a_n)_{n=1}^{\infty} \) of real numbers is a Cauchy sequence if and only if [...].
A: it is convergent
BackOnlyText: The forward implication has been introduced already; it suffices to prove the reverse—that all Cauchy sequences are convergent.
The reverse is not a given: so far, for real sequences, we have delt with steadiness (Cauchy sequences) and closeness between a sequence and a real (convergence); however, we haven't introduced, until now, the idea that a steady sequence must converge to another real.
This theorem along with the closely reated idea of the existance of the supremum sets the reals apart from the rationals. In the language of metric spaces, the theorem asserts that the reals do not contain any 'holes'. The rationals in comparison, do contain holes—rationals can approach something that isn't a rational (and from both sides, hence the idea of a 'hole').
Note on equality
If the 'is' in the statment is considered 'is equal to' (which I think is an important distinction, as equality and entity can be different- two Cauchy sequences can be equal in that they represent the same real, yet they can contain different sequence values), then the Cauchy sequence here, I think, can be considered a Cauchy sequence of reals or rationals. If a sequence of reals converges to a real, then the real to which it converges can be expressed as a Cauchy sequence of rationals.
Source: Tao, Analysis I
Chapter 6, p146

Title: Subsequences, definition
Q: Let \( (a_n)_{n=0}^{\infty} \) and \( (b_n)_{n=0}^{\infty} \) be sequences of reals. We say that \( (b_n)_{n=m}^{\infty} \) is a subsequence of \( (a_n)_{n=m}^{\infty} \) if there exists a function \( f: \mathbb{N} \rightarrow \mathbb{N} \) which is [...] such that
\[ b_n = a_{f(n)} \text{ for all } n \in \mathbb{N} \]
Given how sequences were originally defined in terms of functions, a more explicit definition would be:
Let \( (a_n)_{n=0}^{\infty} \) be the sequence represented by the function \( g: \mathbb{N} \rightarrow \mathbb{R} \). If \( f: \mathbb{N} \rightarrow \mathbb{N} \) is [...], then the sequence defined by [...] is a subsequence of \( (a_n)_{n=0}^{\infty} \).
A: strictly increasing (i.e., \( f(n+1) > f(n) \) for all \( n \in \mathbb{N} \)); strictly increasing; \( g \circ f \)
BackOnlyText: We use 0 as the starting index to make the definition simple.
Lemma 6.6.4
Let \( (a_n)_{n=0}^{\infty} \), \( (b_n)_{n=0}^{\infty} \) and \( (c_n)_{n=0}^{\infty} \) be sequences of real numbers. Then \( (a_n)_{n=0}^{\infty} \) is a subsequence of \( (a_n)_{n=0}^{\infty} \). Furthermore, if \( (b_n)_{n=0}^{\infty} \) is a subsequence of \( (a_n)_{n=0}^{\infty} \) and \( (c_n)_{n=0}^{\infty} \) is a subsequence of \( (b_n)_{n=0}^{\infty} \), then \( (c_n)_{n=0}^{\infty} \) is a subsequence of \( (a_n)_{n=0}^{\infty} \).
Source: p150

Title: Subsequences and limits, proposition
Q: Subsequences and limits
Let \( (a_n)_{n=0}^{\infty} \) be a sequence of real numbers, and let \( L \) be a real number. Then the the following two statements are logically equivalent:
- The sequence \( (a_n)_{n=0}^{\infty} \) converges to \( L \).
- [...]
Subsequences related to limit points
Let \( (a_n)_{n=0}^{\infty} \) be a sequence of real numbers, and let \( L \) be a real number. Then the following two statements are logically equivalent:
- [...]
- There exists a subsequence of \( (a_n)_{n=0}^{\infty} \) which converges to \( L \).
A: Every subsequence of \( (a_n)_{n=0}^{\infty} \) converges to \( L \).; \( L \) is a limit point of \( (a_n)_{n=0}^{\infty} \)
BackOnlyText: These two propositions show a sharp contrast between limits and limit points.
Source: p151

Title: Bolzano-Weierstrass theorem
Q: Let \( (a_n)_{n=0}^{\infty} \) be a bounded sequence (i.e. there exists a real number \( M > 0 \) such that \( |a_k| \le M \text{ for all } k \in N \)). Then there is at least one [...] which [...].
A: subsequence of \( (a_n)_{n=0}^{\infty} \); converges
BackOnlyText: Importance
The Bolzano-Weierstarss theorem says that if a sequence is bounded, then eventually, it has no choice but to converge in some places; it has no "room" to spread out and stop inself from acquiring limit points. This is not the case for unbounded sequences.
In the language of topology, the Bolzano-Weierstrass theorem means that the interval \( \{x \in \mathbb{R}: -M \gt x \gt M \} \) is compact, whereas an unbounded set such as the real line \( \mathbb{R} \) is not compact.
Tao claims that this distinction between compact and non-compact becomes very important in later chapters and that it is of similar importance to the distinction between finite sets and infinite sets.
Proof 1 (Tao)
Proof summary: there must be a limit point, so there must be a convergent subsequence.
Let \( L \) be the limit superior of the sequence \( (a_n)_{n=0}^{\infty} \). Since we have \( -M \le a_k \le M \text{ for all } k \in N \), it follows from the comparison principle that \( -M \ge L \le M \). In particular, \( L \) is a real number (not \( \infty \) or \( -\infty \)). By Proposition 6.4.12(e), \( L \) is thus a limit point of \( (a_n)_{n=0}^{\infty} \) (that proposition simply is this statement). Thus, by Proposition 6.6.6 (subsequences related to limit points), there exists a subsequence of \( (a_n)_{n=0}^{\infty} \) which converges, and it converges to \( L \).
Proof 2 (Abbott)
Abbott has a slightly less formal proof, but one that requires less of a setup compared to Tao's proof (no need to prove that that \( \limsup \) is a limit point, or that for any limit point there is a subsequence that converges to it).
Proof essence: keep splitting the (bounded) set in two closed intervals (one point overlaps), and choosing one of the two sides which has an infinite number of points (at least one of the sides must have an infinite number of points). Repeating this process results in one finding ever-decreasing sized intervals that contain infinite points. If we choose one point from each of these intervals, we can construct a sequence that converges. It converges due to the intersection of nested closed intervals being non-empty.

Title: Monotone bounded sequences converge
Q: Increasing, decreasing and monotone
A sequence \( (a_n)_{n=0}^{\infty} \) is increasing if \( a_n \le a_{n+1} \) for all \( n \in \mathbb{N} \) and decreasing if \( a_n \gt a_{n+1} \) for all \( n \in \mathbb{N} \). A sequence is monotone if it is either increasing or decreasing.
Monotone bounded sequences converge
If a sequence is [...] and [...], then it converges.
The easiest card ever. The proof is on the reverse; can you think of it? The proof is easy and obvious in retrospect, but it uses a property that I originally didn't consider using to solve the problem.
A: monotone; bounded
Source: Abbott, p57
Tao, p138

Title: Square root, expressed as the limit of a sequence
Q: Let \( c \in \mathbb{R} \) be a real. The sequence defined recursively below converges, and it converges to \( \sqrt{c} \).
Let \( x_1 = c \), and let \( x_n \) be defined like so:
[\[ x_{n + 1} = ?(x_n + \; ?) \]]
A: \[ x_{n + 1} = \frac{1}{2}(x_n + \frac{c}{x_n}) \]
BackOnlyText: Proof by induction (any other typical methods) can be used to show that sequences defined recursively like this do (or don't) converge.
Intuition
The outer \( \frac{1}{2} \) factor can be seen as taking an average of the two inner terms. Inspecting the second term, \( \frac{1}{2}(c + \frac{c}{c}) = \frac{c + 1}{2} \), the third, \( \frac{1}{2}(\frac{c+1}{2} + c \frac{2}{c+1}) \), and so on, one can be convinced that this sequences is increasing starting from the third term. In addition, it can be seen that the terms don't surpass \( \sqrt{c} \). While not a exactly convincing as a proof, you can check if a term did surpass \( \sqrt{c} \), then the following term would be less than that term.
Source: Abbott, p54

Title: Convergence of subsequences
Q: Convergence of subsequences
All subsequences of a convergent sequence [what?].
Why is this useful? Two separate uses are listed on the reverse side.
A: converge to the same limit as the original sequence
BackOnlyText: Proof
This proof is just unwrapping then wrapping up the definition of convergence.
Proof. Let \( (a_n)_{n=1}^{\infty} \) be a convergence sequence, converging to \( L \in \mathbb{R} \). Proof by contradiction. Assume that there exists a subsequence \( (a_{f(n)})_{n=1}^{\infty} \), where \(f\) is an increasing function \( f : \mathbb{N} \to \mathbb{N} \), which doesn't converge to \( L \). So there exists a real \( \varepsilon > 0 \) and an \(N \in \mathbb{N} \) such that not all terms \( (a_{f(n)})_{n=N}^{\infty} \) are \( \varepsilon\)-close to \( L \). These terms are all terms within the original sequence, so \( (a_n)_{n=1}^{\infty} \) cannot converge to \( L \).
Surprisingly useful!
The theorem is extremely simple, and the proof too is trivial, yet the theorem is quite useful, almost as a 'trick' for thinking about sequences in a way that helps prove convergence/non-convergence.
Proving divergence
Utilize the power of Modus tollens (denying the consequent). Given an implication \( A \implies B \), proving \( \lnot B \) is sufficient to prove \( \lnot A \). In this way, subsequence convergence is a necessary criterion for a sequences convergence and finding a subsequence that doesn't converge is sufficient to demonstrate that a sequence doesn't converge.
Divergence criterion example
The sequence:
\[ \left( \frac{1}{3}, -\frac{1}{3}, \frac{1}{3}, -\frac{1}{3}, \frac{1}{3}, -\frac{1}{3}, \frac{1}{3}, -\frac{1}{3} ...\right) \]
can be easily shown to be divergent as it has two subsequences which converge to different limits. One converges to \( \frac{1}{3} \):
\[ \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3} ...\right) \]
While the other to \( -\frac{1}{3} \):
\[ \left( -\frac{1}{3}, -\frac{1}{3}, -\frac{1}{3}, -\frac{1}{3} ...\right) \]
Proving convergence
If a convergent sequence contains subsequences which can be related to each other through the algebraic properties of limits, then we can use this relationship, along with the fact that the subsequences have the same limit, to deduce statements about the value of the limit.
Example
Consider the sequence: \( (b^n)_{n=1}^{\infty} \), where \( 0> b > 1 \). We know that:
\[ b > b^2 > b^3 > b^4 > 0 \]
If we consider the sequence \( (b^n b^n)_{n=1}^{\infty} = (b^{2n})_{n=1}^{\infty} \) , we can see that this sequence is a subsequence of \( (b^n)_{n=1}^{\infty} \), and thus, \( (b^{2n})_{n=1}^{\infty} \) and \( (b^{n})_{n=1}^{\infty} \) must have the same limit. Call this limit \( L \in \mathbb{R} \).
In addition, by the algebraic properties of limits, we know that \( \lim_{n \to \infty}b^{2n} =  (\lim_{n \to \infty} b^n ) (\lim_{n \to \infty} b^n ) \). In other words, \( L = L^2 \). But the only real for which this is true is \( L = 0 \). So we have found that \( \lim_{n\to\infty}b^{n} = 0\).
Source: Abbott, p55