Subsequence

Topics: Calculus - Sequence

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(definition)

Let $\{a_n\}\subseteq \mathbb{R}$ be a sequence.

We say that $\{b_n\}$ is a subsequence of $\{a_n\}$ if $\forall s\in \mathbb{N}$, $\exists n\in \mathbb{N}$ such that $b_s=a_n$.

Further more, the order of the sequence $\{a_n\}$ is inherited.

(lemma)

All sequences $\{a_n\}$ contain a monotone subsequence.

Bolzano’s Theorem §

(theorem)

Let $\{a_n{\}}_{n\in \mathbb{N}}$ be a bounded sequence, then $\{a_n\}$ has a convergent subsequence.

Proof

From a specific lemma, we know that $\exists \{b_n{\}}_{n\in \mathbb{N}}$, a monotone subsequence of $\{a_n\}$, that is bounded.

$\{b_n\}$ being a bounded monotone sequence implies it converges.