= Solution
For $K>0$, the <Cauchy-Schwarz inequality> and part a give
$$
\mathbb E\left[Y_n^-\mathbf1_{\{Y_n^->K\}}\right]
\leq\sqrt{\mathbb E[(Y_n^-)^2]}
\sqrt{\mathbb P(Y_n^->K)}
\leq\frac1K.
$$
Thus $(Y_n^-)$ is <uniform integrability>[uniformly integrable]. Combining this with the <weak convergence of random variables> from part b yields convergence of the first moments:
$$
\mathbb E[Y_n^-]\longrightarrow\mathbb E[Y^-].
$$
By symmetry of the standard normal density,
$$
\mathbb E[Y^-]
=\int_0^\infty x\frac{e^{-x^2/2}}{\sqrt{2\pi}}\,dx
=\frac1{\sqrt{2\pi}}.
$$
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