Theorem based on convergence in distribution(or law)

Question

Let $X_n$ be a sequence of integer-valued Random Variables(RVs). Also, let $f_n(k) = P\{X_n = k\}, k = 0,1,2,...$ , be the PMF of $X_n, n = 1,2,... ,$ and $f(k) = P\{X = k\},k =0,1,2,...,$ be the PMF of X. Then prove that

$$f_n(x)\rightarrow f(x) \space\forall\space x \iff X_n\overset{\mbox{L}}{\longrightarrow}X $$

(where L denotes convergence in distribution)

Answer 1

Hint:

If $F$ denotes the CDF of a random variable that takes values in $\{0,1,2,\dots\}\subseteq\mathbb Z$ then:$$F(x)=\sum_{k=0}^{\lfloor x\rfloor}P(X=k)$$

Note that we are dealing with finite sums. If the terms of a finite sum converge then so does the sum itself.

Further $X_n\to X$ in distribution iff $F_n(x)\to F(x)$ for every $x$ at which $F$ is continuous.