= Solution
First take a bounded elementary <predictable process> $\alpha=\sum_jK_j\mathbf1_{(s_j,t_j]}$, with each $K_j$ bounded and $\mathcal F_{s_j}$-measurable and with finite time support. The <Itô integral> is the corresponding finite sum $\sum_jK_j(W_{t_j}-W_{s_j})$. Applying part (a) term by term gives
$$
\mathbb E\left[\phi(W_T)\int_0^\infty\alpha_u\,dW_u\right]=\mathbb E\int_0^T\phi'(W_T)\alpha_u\,du.
$$
Such elementary <predictable processes> are dense among predictable processes in $L^2(d\mathbb P\,du)$. The <Itô isometry> makes the left functional continuous, with bound
$$
\left|\mathbb E\left[\phi(W_T)\int\alpha\,dW\right]\right|\le\|\phi\|_\infty\left(\mathbb E\int_0^\infty\alpha_u^2\,du\right)^{1/2}.
$$
The <Cauchy-Schwarz inequality> makes the right functional continuous, with bound $\|\phi'\|_\infty\sqrt T\left(\mathbb E\int\alpha_u^2du\right)^{1/2}$. Approximation therefore proves \b[the same identity for every allowed predictable $\alpha$]. The integral over the infinite time interval is the $L^2$ limit of its finite-horizon <Itô integrals>.
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