Construct a sequence of random variables

Question

Let $a_n$ be a given sequence of numbers (constant), $X$ be a standard normal random variable. Construct a sequence of random variables $X_n$ such that $X_n \rightarrow X \text{ a.s.}$ and $E(X_n) = a_n$ for all $n$.

I understand that this essentially tells us why dominated-ness is needed in DCT. Now, I tried to construct $X_n$ such that for any $\epsilon >0$ we have $\sum_{n=1}^{\infty}P(|X_n-X|> \epsilon) < \infty$ as this a sufficient condition for almost sure convergence. But, I don't really see how to complete the construction.

Answer 1

Thanks to Dominik in the comments, we can define $X_n = X + 2^n a_n B_n$ where $B_n \sim Ber(\frac{1}{2^n})$. This concludes our construction.