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Let's call $I_1$ and $I_2$ your two integrals respectively. You know that $u_n \to u$ strongly in $L^p$. Because $\Omega$ is bounded, $u_n$ also converges to $u$ strongly in $L^1$. Seemingly, you assume that $f$ is bounded in its two arguments so $$|I_1| \leq \|f\|_{L^{\infty}(\Omega \times \ma...