Increasing and Decreasing Sequence

Topics: Calculus - Sequence

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(definition)

Let $\{a_n{\}}_{n\in \mathbb{N}}$ be a sequence.

We say that this sequence is strictly increasing if for $s>t⟹a_s>a_t$. Similarly, we say that it is decreasing if for $s>t⟹a_s<a_t$.

On the other hand, we say that the sequence is increasing or non-decreasing if for $s>t⟹a_s\ge a_t$. Similarly, we say that it is decreasing or non-increasing if for $s>t⟹a_s\le a_t$.

Increasing or decreasing sequences are called monotone sequences.

Bounds and Convergence §

(theorem)

If $\{a_n\}$ is a bounded monotone sequence, then $\{a_n\}$ converges.

Proof

Without loss of generality, let’s suppose that $\{a_n\}$ is increasing.

Since $\{a_n\}$ is bounded, then $\exists \alpha \in \mathbb{R}$ such that $\alpha =\sup \{x\in \mathbb{R}|x=a_n\}=\sup (A)$.

Let $\epsilon >0$. Since $\alpha$ is the supremum, then $\exists a_N\in A$ such that $\alpha -a_N<\epsilon$.

Since the sequence is increasing, then $\forall m>N$:

$$\begin{array}{rl}{a}_m& >a_n\\ -a_m& <-a_n\\ \alpha -a_m& <\alpha -a_N<\epsilon \\ |\alpha -a_m|& <\epsilon \end{array}$$

With transitivity, we get that:

$$a_m>a_n⟹|\alpha -a_m|<\epsilon \text{ if }m>N$$