# Convergence Distribution and Probability [closed]

Suppose that $$|X_n - Y_n|$$ converges in probability to 0, and that $$X_n$$ converges in distribution to X. Show that $$Y_n$$ converges in distribution to X.

Goal: show $$P(Y_n \le t) \to P(X \le t)$$ [for $$t$$ at which $$F_X(t):=P(X \le t)$$ is continuous].
• $$P(Y_n \le t) = P(Y_n \le t, |X_n - Y_n| > \epsilon) + P(Y_n \le t, |X_n - Y_n| \le \epsilon)$$.
• $$P(Y_n \le t, |X_n - Y_n| > \epsilon) \le P(|X_n - Y_n| > \epsilon)$$. What does the right-hand side converge to?
• $$P(Y_n \le t, |X_n - Y_n| \le \epsilon) \le P(X_n \le t + \epsilon)$$. What does the right-hand side converge to?
• Use the assumption that $$F_X(t)$$ is continuous at $$t$$.