Comparison of limits in distribution Assume we have two sequences of rv. defined on the same probability space $\{X_{n}\}$ and $\{Y_{n}\}$ such that 
$$
X_{n} \stackrel{d}{\to} X
$$
$$
Y_{n} \stackrel{d}{\to} Y
$$
and $P[X_{n} \geq Y_{n}] \to 1$ as $n\to\infty$. 
Then what can be said about $X$ and $Y$? Is it true that $X\geq_{s}Y$? Here $\geq_{s}$ means stochastically greater.
 A: Assume for simplicity of notations that $X$ and $Y$ are defined on the same probability space. Let $t$ be a continuity point of the cumulative distribution function of $X$ and $Y$. Then 
\begin{align}
 \mathbb P \left\{Y \gt t\right\}  &= \lim_{n\to+ \infty } \mathbb P \left\{Y_n \gt t\right\}    \\
& \leqslant \limsup_{n\to+ \infty } \mathbb P \left\{X_n \lt Y_n\right\} 
+  \mathbb P \left( \left\{Y_n \gt t\right\} \cap  \left\{X_n \geqslant Y_n\right\}\right)\\
&  \leqslant \limsup_{n\to+ \infty } \mathbb P \left\{X_n \lt Y_n\right\} 
+  \limsup_{n\to+ \infty } \mathbb P  \left\{X_n \gt t\right\}   \\
&=\mathbb P \left\{X \gt t\right\}.
\end{align}
Since the functions $g_1\colon t\mapsto   \mathbb P \left\{X \gt t\right\}$ and $g_1\colon t\mapsto \mathbb P \left\{Y \gt t\right\}$ are right-continuous and the discontinuity points of $g_1-g_2$ is at most countable, the equality 
 $$ \mathbb P \left\{Y \gt t\right\} \leqslant  \mathbb P \left\{X \gt t\right\}  $$
holds for any real number $t$.
