Convergence of sequence of rv in distribution and pointwise

Question

Let $X \sim \mathcal{N}\left(\mu,\sigma^2\right)$ and define a sequence of random variables $$ M_n = \begin{cases} X, & n \text{ is even}\\ 3-X, & n \text{ is odd} \end{cases} $$ For which values of $\mu$ and $\sigma^2$ do $M_n$ converge in distribution? For these values of $\mu$ and $\sigma^2$, is there a.s. convergence?

Answer 1

HINT

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1. What is the actual distribution of $3-X$?
2. Can you set $\mu$ and $\sigma^2$ cleverly so that $X$ and $3-X$ have the same distribution?
3. Now that the same distribution has been achieved, is there a.s. convergence?