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Fix $ϵ>0$ and use the indicator random variables $Y_n=1_{\{|X_n-X|>ϵ\}}$. Almost sure convergence gives ${\mathrm{limsup}}_n{Y}_n=0$ almost surely, while $0\le Y_n\le 1$. The Reverse Fatou lemma states that for random variables bounded above by an integrable random variable,It follows thatSince $ϵ$ was arbitrary, this proves convergence in probability. The integrable upper bound is essential to the application of the Reverse Fatou lemma.