Math and science::Analysis::Tao::06. Limits of sequences
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 IChapter 6, p131