= Solution
The independent standard <Gaussian random variables> have mean zero and variance one. For $m>n$, the mixed products have expectation zero, giving
$$
\|Y_m-Y_n\|_2^2=\mathbb E\left(\sum_{j=n+1}^m\alpha_jX_j\right)^2=\sum_{j=n+1}^m\alpha_j^2\longrightarrow0
$$
uniformly over $m>n$ as $n\to\infty$. Thus $(Y_n)$ is Cauchy in the <L2 space>. Its completeness, proved in (i), supplies
$$
\boxed{\exists Y\in L^2:\quad\|Y_n-Y\|_2\longrightarrow0.}
$$
Moreover $L^2$ convergence implies convergence of the means and second moments: $\mathbb EY=0$ and $\mathbb EY^2=\lim_n\sum_{j\leq n}\alpha_j^2=\sigma^2$.
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