Calculating function convergence
1. A symbolic expression
Consider the very simple, somewhat informal, expression:
2. The exact expression
Expressed exactly according to the wording of the definition of function convergence, the above expression represents the object:
The real to which the function \( f: \mathbb{R} \to \mathbb{R} \), defined by \( f(x) := c \), [...].
I would argue that symbolic form is only exact when \( c \) is replaced with the explicit function:
The definition is paired with a proof that such a number is unique, so the use of 'the' instead of 'a' is justified.
3. Unwrapping the definition of function convergence.
If we expand the meaning of "converges to <\( i \)>, at <\( x_0 \)> on <\( E \)>" according to the definition of convergence, the object becomes:
The real to which the function \( f: \mathbb{R} \to \mathbb{R} \), defined by \( f(x) := c \), when restricted to \( \mathbb{R} \), is ε-close to [...] for [...].
"when" was introduced into the wording, as without it it was not clear that "is ε-close to near \( x_0 \)" means "is ε-close to <the object> near \( x_0 \)".
4. We can check, but not compute
The statement in 3 allows us to check if a certain real \( L \) matches the criteria to be able to be equated with the expression in 1. This isn't that useful, as you kind of need to know what the value is ahead of time. For example, we can check that the real \( c \) matches the criteria:
For any \( \varepsilon > 0 \), we can find a \( \delta > 0 \) such that when \( f \), restricted to \( \mathbb{R} \), is further restricted to \( \{x \in \mathbb{R} : |x - x_0| < \delta \} \), the function is ε-close to \( c \). For example, let \( \varepsilon > 0 \) be arbitary. Choose \( \delta = 1 \), and consider \( f \) restricted to \( \{x \in \mathbb{R} : |x - x_0| < 1 \} = (x_0 - 1, x_0 + 1) \). For any \( x \) in this set we have \( |f(x) - c| = |c - c| = 0 < \varepsilon \). Thus, \( c \) meets the criteria to be able to equate it to the expression from 1.
5. Using Proposition 9.3.9
What an ordeal... This is why Prop. 9.3.9 is so important. Repeated here it is:
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( E \) be a subset of \( X \), let \( x_0 \) be an adherent point of \( E \), let \( L \) be a real, and let \( \varepsilon > 0 \) be a real. Then the following two statements are logically equivalent:
- \( f \) converges to \( L \) at \( a_0 \) in \( E \).
- For every sequence \( (a_n)_{n=0}^{\infty} \) which consists entirely of elements of \( E \) and converges to \( x_0 \), the sequence \( (f(a_n))_{n=0}^{\infty} \) converges to \( L \).
Using this proposition, we can calculate the value of the expression in 1. This is because we are able to express everything in terms of limits of sequences, for which we have already developed a wealth of propositions covering equality, such as Theorem 6.1.19. Using these theorems, we don't need to rederive results like I did above for the constant function \( f(x) := c \).
6. Utilize sequence limit laws
Taking our expression from 1, Prop. 9.3.9 allows us to replace it with the expression:
The object to which any sequence \( (f(a_n))_{n=0}^{\infty} \) converges to as long as the sequence \( (a_n)_{n=0}^{\infty} \) converges to \( x_0 \).
\( (f(a_n))_{n=0}^{\infty} = (c)_{n=0}^{\infty} \), by definition of \( f \). Finally, \( (c)_{n=0}^{\infty} = c \), by Theorem 6.1.19. So:
And we can ignore the requirements on sequence \( (a_n)_{n=0}^{\infty} \), as \( f(a_n) \) is independent of \( a_n \)