Proof of Convergence in Distribution and Limsup I'm currently using 'Adventure in Stochastic Processes' for self-study. Here's the link. This is the part I don't understand:
Letting $\left[n\rightarrow \infty\right]$, we get $\limsup_{n\rightarrow \infty} \mid P_n(s)-P_o(s)) \mid \leq \epsilon$.
I have never learned about limsup, so I'm confused what this means. Why not just use limit?


Why not just use limit?

Because, as is often the case, one does not know yet whether a limit exists. Limsups always exist hence they are useful to show that limits exist and to determine them. 
To wit, a sequence $(x_n)$ converges to $\ell$ if and only if $\limsup\limits_{n\to\infty}|x_n-\ell|=0$ if and only if, for every $\varepsilon\gt0$, $\limsup\limits_{n\to\infty}|x_n-\ell|\leqslant\varepsilon$.
