= Convergence in probability implies convergence in distribution
{title2=$X_n\xrightarrow{p}X\Longrightarrow X_n\xrightarrow{d}X$}
For real <random variables> on one <probability space>, fix a <continuity> point $x$ of the <cumulative distribution function> $F_X$. For every $\varepsilon>0$, inclusions of <events> give
$$
F_X(x-\varepsilon)-\mathbb P(|X_n-X|>\varepsilon)\leq F_{X_n}(x)\leq F_X(x+\varepsilon)+\mathbb P(|X_n-X|>\varepsilon).
$$
Take the lower and upper <limits> as $n\to\infty$, using <convergence in probability>, and then let $\varepsilon\downarrow0$. <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>.
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