Let $f_{i}\colon X\rightarrow \mathbb{R}$ be convex, Lipshitz and positive (X - reflexive Banach space) for $i=1,2$. One may easily prove that $f_{i}$ is weak lower semicontinuous. How to prove the following statement:
$lim\, inf\, f_1(x_{n})[f_{2}(x_{n})-f_{2}(x)]\geq0$, where $\{x_{n}\}$ converges weakly to some $x$?
We can write $lim\, inf\, f_1(x_{n})[f_{2}(x_{n})-f_{2}(x)]\geq\lim\, inf\, f_1(x_{n})\cdot lim\, inf\, [f_{2}(x_{n})-f_{2}(x)]$ (it would solve the problem)
but
$(f_{2}(x_{n})-f_{2}(x))$ has to be positive and I don't know that. Maybe convexity has something to do with the proof?