Let $F : H → H$ be continuous function on a Hilbert space H such that $<F (x) − F (y), x − y> ≥ 0, ∀x, y ∈ H × H$.
Let $x_n \rightharpoonup x$ with $x_n$ a sequence in H such that $F (x_n ) \to l\in H$ when $n\to \infty$.
Prove that $∀y ∈ H, <l − F (y), x − y> ≥ 0 $.
Here we have $\forall n\in \Bbb n: <F (x_n) − F (y), x_n − y> ≥ 0$ and I am not sure if we can just make $n\to \infty$ to find the requested inequality? It does not seem that simple to me.
Thank you for your help.