Convergence in probability implies convergence in distribution ($X_n\stackrel{p}{\to }X⟹X_n\stackrel{d}{\to }X$)

For real random variables on one probability space, fix a continuity point $x$ of the cumulative distribution function $F_X$. For every $\epsilon >0$, inclusions of events giveTake the lower and upper limits as $n\to \infty$, using convergence in probability, and then let $\epsilon \downarrow 0$. Continuity at $x$ makes both bounds $F_X(x)$, proving convergence in distribution. The two modes of convergence have different information: convergence in probability concerns the coupling on one probability space, whereas convergence in distribution concerns only the marginal probability distributions.