Past exam of the options II course of the University of Oxford / 2012 / M / M4 / c

Assume convergence in probability and fix $\epsilon >0$. On the event $|Y_n-Y|\le \epsilon$, $Y\le x-\epsilon$ implies $Y_n\le x$, while $Y_n\le x$ implies $Y\le x+\epsilon$. HenceTaking lower and upper limits in $n$ removes the error probabilities. If $x$ is a continuity point of $F_Y$, letting $\epsilon \downarrow 0$ squeezes both bounds to $F_Y(x)$. Thus