# Convergence of Frechet Derivative Functional related to Palais-Smale Sequence

Let $$\phi : H_{0}^{1}(\Omega)\to\mathbb{R}$$ be an energy functional for bounded domain $$\Omega$$ such that $$\phi'$$ be its Frechet derivative, and there exists a bounded sequence $$\{u_{n}\}_{n\in\mathbb{N}}$$ bounded in $$H_{0}^{1}(\Omega)$$ such that $$\phi(u_{n})\to d$$ and $$\phi'(u_{n})\to 0$$ as $$n\to\infty$$. Assume that $$u_{n}\to u$$ weakly in $$H_{0}^{1}(\Omega)$$ and define $$\langle\,\cdot\, , \,\cdot\,\rangle$$ as the inner dot product in $$H_{0}^{1}(\Omega)$$.

Claim : $$\langle\phi'(u_{n})-\phi'(u),u_{n}-u\rangle\to 0$$ as $$n\to\infty$$.

My question is how to show that claim. My attempt so far is to separate it into : $$\langle\phi'(u_{n}),u_{n}-u\rangle - \langle\phi'(u),u_{n}-u\rangle$$ The second term tends to zero as $$n\to\infty$$ by weak convergence of $$\{u_{n}\}_{n\in\mathbb{N}}$$ but I am not sure how to show that the first term tends to zero since $$u_{n}-u$$ still depends on $$n$$ so I do not know whether I could use the assumption that $$\phi'(u_{n})\to0$$ as $$n\to\infty$$.

Any help is much appreciated! Thank you very much!