Here is a theorem on my course:
If $X_n \to^{\mathbb{P}} X$ (in probability) and $\sum\limits_{n=1}^{\infty}\mathbb{P}[|X_n-X|>\epsilon] < \infty$ for all $\epsilon > 0 $ then $X_n \to X$ almost surely.
I have that $X_n \to X$ almost surely if and only if $\mathbb{P}[|X_n-X|>\epsilon$ i.o] = $0$ for all $\epsilon > 0 $
Hence by Borel Cantelli 1
$\sum\limits_{n=1}^{\infty}\mathbb{P}[|X_n-X|>\epsilon] < \infty$ for all $\epsilon > 0 \implies \mathbb{P}[|X_n-X| >\epsilon$ i.o] = $0$ for all $\epsilon > 0 \implies X_n \to X$ a.s
Why do I need the requirement that $X_n \to^{\mathbb{P}} X$ in probability ?
Thanks lots!