# Prove bi-directional relationship between convergence in distribution and convergence of probability mass functions

Let $$X$$ be a random variable that is positive and integer-valued.

Let $$X_1, X_2, ...$$ also be random variables that are positive and integer-valued.

Prove that $$X_n$$ converges in distribution to $$X$$ if and only if

$$\lim_{n\to\infty} P(X_n = k) = P(X = k)$$

for every integer $$k$$.

My attempt:

I know that convergence in distribution for this problem means that

$$\lim_{n\to\infty} P(X_n \leq k) = P(X \leq k)$$

I don't know how this leads to the desired result. I also don't know how to prove the reverse implication.

• Hint: $P(X_n=k) = P(X_n\le k) - P(X_n\le k-1).$ – whuber Dec 1 '18 at 22:01
• Thanks @whuber. How do I apply that as $n$ goes to infinity? – MSE Dec 1 '18 at 22:29
• The limit of the difference of two sequences is the difference of their limits (QED). – whuber Dec 1 '18 at 23:31