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Math and science::Analysis::Tao::06. Limits of sequences
Subsequences, definition
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 strictly increasing (i.e., \( f(n+1) > f(n) \) for all \( n \in \mathbb{N} \)) 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 strictly increasing, then the sequence defined by \( g \circ f \) is a subsequence of \( (a_n)_{n=0}^{\infty} \).
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
