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Suppoe $X$ is a Banach spaces and $G\subset X$ is a convex open set. Let $\phi:G\rightarrow \mathbb{R}$ be a $C^1$ function and assume that $\phi'$ is a bounded and peseudo-monotone map (see here for a definiton of pseudo-monotone).

We say that $\phi$ is weakly sequentially lower semicontinuos (WSLSC) in $G$ if for every sequence $x_n$ in $G$ which converges weakly to $x\in G$ we have that $$\phi(x)\leq\liminf \phi(x_n)$$

How can i show that $\phi$ is WSLSC?

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