On $\sum_{k=1}^\infty ( \lim_{n \to \infty} P(X_n =k))$ when $\lim_{n \to \infty} P(X_n =k)$ exists

Question

Let $\{X_n \}$ be a sequence of independent random variables taking only positive integer values. If for every integer $k \ge1$ , we have that $\lim_{n \to \infty} P(X_n =k)$ exists, then is it true that $\sum_{k=1}^\infty ( \lim_{n \to \infty} P(X_n =k))$ is convergent ?

Answer 1

Yes, and in fact, the sum id s always $\leq 1$. This is a consequence of Fatou's Lemms which shows that $\sum_k \lim_{n\to \infty} P(X_n=k) \leq \lim \inf_{n\to \infty} \sum_k P(X_n=k)=1$.

Alternately you can argue that each finite sum in the series is bounded by $1$. This avoids Fatou's Lemma.