$\liminf $ and $\sup $ inequality

Question

It is clear to me that $\liminf X_n \le \limsup X_n$ but every time that I see that $\liminf X_n \le \sup X_n$ it sound to me not so obvious awkward. Could someone help me to understand this inequality?$ $ $ $

Answer 1

Isn't it clear from the definition? Limsup is less than equal to sup.

Proof:

The definition of limsup is the following: Inf{sup$(T_n)$} where $T_n$ is the $n$th tail of the sequence $x_n$. Clearly, sup$(T_n)$ is less than equal to sup($x_n$). Now, just take infimum over $n$. Then we get that limsup$x_n$ $\leq$ sup$x_n$.

By the way, the definition of $T_n$ is {$x_k| k\geq n$}.

Answer 2

Hint

As $\limsup X_n \le \sup X_n$, this second inequality $\liminf X_n \le \sup X_n$ should be obvious to you if $\liminf X_n \le \limsup X_n$ is.

So you're left to prove and think why $\limsup X_n \le \sup X_n$.

Answer 3

Hint:

$\sup X_n$ is an upperbound of the set $\{X_n\mid n\in\mathbb N\}$ which means that for every $k\in\mathbb N$ we have $X_k\leq\sup X_n$.

Answer 4

By definition, $$\limsup X_n=\lim_k\Bigl(\sup_{n\ge k} X_n\Bigr), $$ and the sequence $(S_k)=\bigl(\sup_{n\ge k} X_n\bigr)_k$ is non-increasing, so $$\liminf X_n\le\limsup X_n\le S_1=\sup X_n.$$