I'm trying to check my knowledge on th enotion of almost sure convergence. I've cooked up this example:
Let $X_1$ be a Bernoulli random variable with paramter $\frac{1}{2}$. Now consider the sequence of random variables $X_1, X_2, X_3, \ldots$, where $X_n = X_{n-1}$ for $n>1$.
Is it true that $X_n$ converges to $X$ almost surely, where $X$ is also a Bernoulli random variable with paramter $\frac{1}{2}$?
For each $\omega$, $\lim_{n\to\infty} X_n(\omega)=X_1(\omega)$, so $X_n$ indeed converges surely to $X_1$. However, I can choose $X=1-X_1$ (note that $X$ is still Bernoulli with parameter $1/2$), and so it is not true that necessarily $X_n\to X$ almost surely.