= Solution
Multiply the representation in part (c) by $\int_0^\infty\alpha_u\,dW_u$ and take expectations. This <Itô integral> has mean zero, and the bilinear form of the <Itô isometry> gives
$$
\mathbb E\left[\phi(W_T)\int_0^\infty\alpha_u\,dW_u\right]=\mathbb E\int_0^\infty\beta_u\alpha_u\,du.
$$
Equating this with part (b), and writing both ordinary integrals with the same time variable, yields
$$
\boxed{\mathbb E\int_0^\infty\left(\beta_u-\phi'(W_T)\mathbf1_{\{u\le T\}}\right)\alpha_u\,du=0.}
$$
All terms are integrable by the <Cauchy-Schwarz inequality>, the assumed square integrability of $\alpha$, and boundedness of $\phi'$. This is an orthogonality statement against <predictable processes>; its second term need not itself be predictable.
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