A sequence of random variables $X_n$ converges pointwise to a weighted sum of two other sequences of random variables $Y_n$ and $Z_n$.
$~~~\lim_{n\to\infty} |X_n - (a Y_n + b Z_n)|=0$
In addition, $Y_n$ and $Z_n$ converge in distribution to independent normal distributions.
$~~~Y_n\stackrel{d}{\to}\mathcal{N}(0,1)$, $~~~~~~~~~Z_n\stackrel{d}{\to}\mathcal{N}(0,1)$.
as $n\to\infty$. Is it true that
$~~~X_n\stackrel{d}{\to}\mathcal{N}(0,a^2+b^2)$
And if yes, what would be the shortest argument? It seems to me that Slutsky's theorem or the continuous mapping theorem can not be directly applied.
Take $Y_n=Z_n=N\sim\mathcal{N}(0,1)$, and let $Y$ and $Z$ be independent standard normal random variables. Then $Y_n\xrightarrow{d}Y$, $Z_n\xrightarrow{d}Z$, and $Y+Z\sim\mathcal{N}(0,a^2+b^2)$. However, $$ X_n=(a+b)N\xrightarrow{d}\mathcal{N}(0, (a+b)^2). $$
The result holds, however, when $X_n$ and $Y_n$ converge jointly to $\mathcal{N}(0,I_2)$.