13 - Monotone Sequences Theorem

Monotone Sequences Theorem

Let $(a_n)$ be a real sequence

1. If $(a_n)$ is increasing and bounded above, then $(a_n)$ converges
2. If $(a_n)$ is decreasing and bounded below, then $(a_n)$ converges

TLDR: Bounded monotone seqence converges

Proof

1. Assume that $(a_n)$ is increasing and bounded above
   Define set $S:=\{a_n:n\ge 1\}$ then it is non-empty and bounded above
   By Completeness Axiom then $\sup S$ exists

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Take $ϵ>0$
By the Approximation Property, there is $N$ such that

$$\sup S-ϵ<a_N\le \sup S$$

As $(a_n)$ is increasing then for $n\ge N$ then

$$\sup S-ϵ<a_N\le a_n\le \sup S$$

Hence

$$|a_n-\sup S|<ϵ$$

So $(a_n)$ converges, and $a_n\to \sup S$ as $n\to \infty$

2. If $(a_n)$ is decreasing and bounded below, then $(-a_n)$ is increasing and bounded above so $(1)$