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

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