Let $X_n$ be a sequence of random variables with $X_n \sim \mathscr{N}(m_k,\sigma^2_k)$.
a) Assume that $m_k \rightarrow m$ and $\sigma^2_k \rightarrow \sigma^2 $. Show that $X_n \rightarrow^d \mathscr{N}(m,\sigma^2)$.
b) Let $X$ be a random variable such that $X_n \rightarrow^d X$. Show that $X \sim \mathscr{N}(m,\sigma^2)$ for some $m$ and $\sigma^2$ such that $m_k \rightarrow m$ and $\sigma^2_k \rightarrow \sigma^2$.
This seems to be a really basic exercise. My idea is to show the pointwise convergence of the characteristic functions and to deduce by Levy's continuity theorem both statements which would be a 1 or 2 line proof. Is this correct? I am wondering about the splitted task as my idea is the same for both a) and b).
Thanks in advance!
Your idea is correct for part (a).
For part (b), the only information you have is that the sequence converges in distribution to some random variable. From this alone, you need to prove (not assume) that the sequences $m_k$ and $\sigma_k^2$ converge and that ultimately the limiting distribution is Gaussian. Using characteristic functions will still work, but it is a little more work than the outline you have provided for (a).