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?
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 $\varepsilon\gt0$, $\limsup\limits_{n\to\infty}|x_n-\ell|\leqslant\varepsilon$.