Proof of liminf subset limsup

Question

I am sure this is fairly simple, but I am just not getting it.

I am not sure how to show that $\liminf_n→_∞ A_n \ $ ⊆ $\limsup_n→_∞ A_n \ $

I guess I am not understanding lim inf and lim sup properly.

Answer 1

Let $x \in \liminf_n A_n$ then exists $N \in \mathbb{N}$ such that $x \in A_n$ forall $n \geq N$

Thus $x$ lies into infinitely many $A_n$'s

proving that $x \in \limsup_n A_n$

Answer 2

$$\lim \inf A_n=\cup_{n=1}^{\infty}\cap_{m\geq n}A_m.$$ $$\lim \sup A_n=\cap_{n=1}^{\infty}\cup_{m\geq n}A_m.$$ Verify that $(\;x\in \lim \inf A_n\;) \iff (\;\{n:x\not \in A_n\}$ is finite $\;$).

Verify that $(\;x\in \lim \sup A_n\;) \iff (\;\{n:x\in A_n\}$ is infinite $\;$).

From these, verify that $x\in \lim \inf A_n\implies x\in \lim \sup A_n.$