Q:

Why $\lim\sup_{n\rightarrow\infty} f_n(x)=\inf_k\{\sup_{n\geq k}f_n\}$

Why 
$$\lim\sup_{n\rightarrow\infty} f_n(x)=\inf_k\{\sup_{n\geq k}f_n\}$$
Somebody thinks it is obviously,.but I can't  get it. 
Hope your nice answer. 

A:

Let $a_k= \{sup_ {n\geq k} f_n\}$. Verify that $a_{k+1} \leq a_k$. For any decreasing sequence $(a_k)$ we have $\lim a_n$ and $\inf a_k$ are the same.  If you change $\inf$ on RHS to $\lim $ you get LHS.