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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 [...] 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 \( (a_n)_{n=0}^{\infty} \).