Monotone_Convergence_Theorem

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Definition

If a sequence is monotone and bounded, then it converges.

More precisely:

• If $(a_n)$ is increasing and bounded above, then $\lim a_n = \sup\{a_n : n \in \mathbb{N}\}$
• If $(a_n)$ is decreasing and bounded below, then $\lim a_n = \inf\{a_n : n \in \mathbb{N}\}$

Proof

Let $(a_n)$ be increasing and bounded above. Let $s = \sup\{a_n\}$, which exists by the axiom_of_completeness.

Given $\epsilon > 0$, by the definition of supremum, there exists $N$ such that $s - \epsilon < a_N \leq s$.

Since $(a_n)$ is increasing, for all $n \geq N$:

$$ s - \epsilon < a_N \leq a_n \leq s < s + \epsilon $$

Thus $|a_n - s| < \epsilon$ for $n \geq N$, so $a_n \to s$.