Non-decreasing sequence of random variable convergence in probability implies it also converges almost surely. The problem stated as follow:

Suppose $X_1 \leq X_2 \leq \cdots$ and $X_n \xrightarrow[]{p} X$. Show that $X_n \to X$ a.s.

I'm think about may be use the continuity of probability measure, but I don't know if that's correct.

You don't need to use the subsequence argument.  For each $\omega$, $\{X_n(\omega)\}$ is an increasing sequence of real numbers, and so it has a limit $Y(\omega) \in (\infty, +\infty]$.  Being a pointwise limit of measurable functions, $Y$ is measurable, i.e. a random variable.  It remains to show that $Y = X$ a.s.  But since $X_n \to Y$ pointwise, we also have $X_n \to Y$ in probability and limits in probability are unique up to null sets, so indeed $Y=X$ a.s.
