$X_1,X_2,...$ are i.i.d. RVs. Now if for a fixed $a$ $$\text{Pr}(X_i=a)>0$$ holds, does $$\sum\limits_{i=1}^n 1(X_i=a) \to \infty$$ Pr-a.s. for $n\to\infty$ hold ? ($1$ being the indicator function)
The Law of Large Numbers gives $$\frac1n \sum\limits_{i=1}^n 1(X_i=a) \to \text{Pr}(X_i=a)$$ Pr-a.s. for $n\to\infty$.
The events $X_i=a$ are independent and $\sum_i P(X_i=a)=P(X_1=a)\sum_i 1 = \infty$.
By Borel-Cantelli theorem, $P(\limsup_i X_i=a)=1$, which implies that $P.$a.s, $\sum_{i=1}^n 1(X_i=a)\to \infty$.
(Since the summands are either $0$ or $1$, the sum diverges iff there are infinitely many 1's)