$X$ be a locally convex topological vector space. $\{x_n\}$ converge weakly to $x$ iff $x$ in the closed convex hull of every subsequence of $\{x_n\}$.
I'm able to show the only if part that is if $x_n\xrightarrow{w} x$ then $\{x\}=\cap K_n$ where $K_n= \bar{co}\{x_{n,i}\}_i$, i.e. closed convex hull of subsequence.
How to show the if part?