= Solution
Conditionally on $Y_1,\ldots,Y_m$, the <cumulative distribution function> of $X^*$ is the weighted empirical distribution
$$
F_m^*(t)=\mathbb P(X^*\leq t\mid Y_1,\ldots,Y_m)
=\frac{m^{-1}\sum_{i=1}^m w(Y_i)\mathbf1_{\{Y_i\leq t\}}}
{m^{-1}\sum_{i=1}^m w(Y_i)},
\qquad w(y)=\frac{f(y)}{h(y)}.
$$
For fixed $t$, the <strong law of large numbers> gives, almost surely,
$$
\frac1m\sum_{i=1}^m w(Y_i)\mathbf1_{\{Y_i\leq t\}}
\longrightarrow
\int_{(-\infty,t]}f(y)\,dy=F(t),
$$
while
$$
\frac1m\sum_{i=1}^m w(Y_i)\longrightarrow\int f(y)\,dy=1.
$$
The ratio therefore converges almost surely to $F(t)$. This is precisely the claimed <convergence in distribution> of the <self-normalized importance sampling> resample.
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