3.3 Monotone Sequences

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Summary

Definition

• Monotone sequences

Theorem

• $X$ sequence of real numbers

1. $X$ monotone →
   • convergent ←> bounded
   • increasing, bounded above → $\lim X=\sup \{x_n:n\in \mathbb{N}\}$
   • decreasing, bounded below → $\lim X=\inf \{x_n:n\in \mathbb{N}\}$

3.3.1 Definition of monotone sequences

Let $(x_n)$ be a sequence of real numbers. $(x_n)$ is increasing if it satisfies: $x_1\le x_2\le \cdots \le x_n\le x_{n+1}\le \dots$ $(x_n)$ is decreasing if it satisfies: $x_1\ge x_2\ge \cdots \ge x_n\ge x_{n+1}\ge \dots$ $(x_n)$ is monotone if it’s either increasing or decreasing.

3.3.2 Theorem: Monotone convergence

A monotone sequence of real numbers is

convergent $\leftrightarrow$ bounded.

Furthermore:

1. If $(x_n)$ is an increasing sequence and bounded above, then $\lim (x_n)=\sup \{x_n:n\in \mathbb{N}\}$
2. If $(y_n)$ is an decreasing sequence and bounded below then $\lim (y_n)=\inf \{y_n:n\in \mathbb{N}\}$

Intuition

Note that a monotone sequence means the sequence is either ALWAYS increasing or decreasing.

If it is convergent, that means there must be a point where the monotonic sequence STOPS increasing or decreasing (converges to the limit).

If it is bounded (bounded BOTH above and below), it must also mean that there is a point where the monotonic sequence STOPS increasing or decreasing (“restricted” by the bound).