Let $X$ be a Hilbert space and $(x_{n})\subset X$ be a infimal sequance i.e., $\lim F(x_{n})=\inf_{x\in X}(F(x))$, where $F\colon X\rightarrow \mathbb{R}\cup \{\infty\}$ is let's say convex and l.s.c. If I consider a subsequance $(x_{n_{k}})$, is it still infimal sequance?
What about $(x_{n_{p}})$ and $x_{n_{p}}\rightarrow \overline{x}$ weakly. Is $(x_{n_{p}})$ an infimal sequance?
If your definition of 'infimal sequence' is $F(x_n) \to \inf_{x \in X}(F(x))$, a subsequence of an infimal sequence is still infimal. This follows from simple real analysis: a subsequence of a convergent sequence in $\mathbb{R}$ is convergent and has the same limit. Apply this statement to the sequence $F\{(x_n)\}_{n \in \mathbb N}$.