I thought of this example in hopes of helping me understand almost sure convergence a little better. So, if you could add any additional (relevant) details in your response I would greatly appreciate it.
By definition, almost sure convergence requires:
\begin{align} P(\lim_{n\to \infty}{X_n = a}) = 1 \end{align} Where $(X_n)$ is a sequence of IId random variables that converges to some real number $a$.
Let us define the $X_n$ as Bernoulli Random Variables: \begin{align} X_n = \begin{cases} 1 & \text{with probability $\frac{1}{n}$}\\ 0 & \text{with probability $1 - \frac{1}{n}$} \end{cases} \end{align}
So, as $n \to \infty \mathbb{P}(X_n = 1) = 0$ and $\mathbb{P}(X_n = 0) = 1$. Does this imply that $X_n \to 0$ almost surely?
Thank you!