= Solution
A finite linear combination of independent <Gaussian random variables> is Gaussian, so
$$
\sum_{k=1}^na_kY_k\sim N(0,v_n),
\qquad
v_n=\sum_{k=1}^na_k^2.
$$
Since $v_n\to v:=\sum_{k\geq1}a_k^2<\infty$, the <characteristic functions> converge pointwise:
$$
\exp\left(-\frac12v_nt^2\right)\longrightarrow\exp\left(-\frac12vt^2\right).
$$
The <Lévy continuity theorem> yields convergence in distribution to
$$
\boxed{N\left(0,\sum_{k\geq1}a_k^2\right)}.
$$
On the given common probability space, the same variance calculation actually proves convergence in $L^2$ as well.
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