Im looking at "A Master Dominated Convergence Theorem" Proposition 11.11 from Gordan Žitkovic's Probability Theory notes (its on page 145) and the link to his notes can be found here: https://www.ma.utexas.edu/users/gordanz/notes/theory_of_probability_I.pdf
I am stuck on the last part of his proof where he says that 
I know that in order to apply the bounded convergence theorem, we need $\Psi_M\left(\left\vert X_n\right\vert^p\right) \overset{a.s.}\longrightarrow \Psi_M(\left\vert X\right\vert^p)$. But we only have that $\Psi_M(X_n)\overset{\mathbb{P}}\longrightarrow \Psi_M(X)$ by the continuous mapping theorem...and I am not too sure how to get to the almost sure convergence given we only have the convergence in probability! Thanks for the help in advanced.