Existence of lim sup/lim inf for bounded positive sequences

Question

Is it true that a sequence that is non-negative and bounded always have a convergent lim sup or lim inf ?

Answer 1

I am going to show that for any sequence $(x_n)_{n\in\mathbb{N}}$ with $-C\leq x_n \leq C$ for all $n\in\mathbb{N},$ we have that $\limsup\limits_{n\to\infty}x_n$ exists and is finite. First note that by definition $$\limsup\limits_{n\to\infty}x_n=\lim\limits_{n\to\infty}\left(\sup\limits_{m\geq n}x_m\right).$$ Consider the sequence $$(y_n)_{n\in\mathbb{N}}:=\left(\sup\limits_{m\geq n}x_m\right)_{n\in\mathbb{N}}.$$ Then obviously, $y_n\geq y_{n+1},$ so that the sequence $(y_n)_{n\in\mathbb{N}}$ is monotone decreasing. Moreover, since $x_n\geq -C$ for all $n,$ $(y_n)_{n\in\mathbb{N}}$ is bounded from below, and so it must converge by https://en.wikipedia.org/wiki/Monotone_convergence_theorem . You can proceed similarly for the $\liminf$.