Math - limsup and liminf

Preparations: Properties of Monotone Sequences

Theorem A monotone sequence of real numbers converges if and only if it is bounded. Moreover,

1. If $\{x_n\}$ is monotone increasing and bounded, then $\lim_{n \rightarrow \infty} x_n = \sup\{x_n:n \in \mathbb{N}\}.$
2. If $\{x_n\}$ is monotone decreasing and bounded, then $\lim_{n \rightarrow \infty} x_n = \inf\{x_n:n \in \mathbb{N}\}.$
3. If $\{x_n\}$ is monotone increasing and unbounded, then $\lim_{n \rightarrow \infty} x_n = \infty.$
4. If $\{x_n\}$ is monotone decreasing and unbounded, then
5. $$\lim_{n \rightarrow \infty} x_n = -\infty.$$

Definitions

Definition Let $\{x_n\}$ be a sequence of real numbers, let $y_n = \sup\{x_k:k \geq n\}$, and $z_n = \inf\{x_k:k \geq n\}$, then $\limsup_{n \rightarrow \infty}x_n = \lim_{n \rightarrow \infty}y_n = \inf_{n \in \mathbb{N}}\{\sup_{k \geq n}x_k\}.$ and $\liminf_{n \rightarrow \infty}x_n = \lim_{n \rightarrow \infty}z_n = \sup_{n \in \mathbb{N}}\{\inf_{k \geq n}x_k\}.$

Proposition Every bounded sequence has $\limsup$ and $\liminf$, and the sequence converges if and only if $\limsup = \liminf$.