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?