Let $X=C[0,1]$ and $x,x_{n}\in X,\forall n\in\mathbb{N}.$ Suppose $\forall t\in[0,1],\ x_{n}(t)\rightarrow x(t)$ and $\sup_{n}\left\Vert x_{n}\right\Vert <\infty$. Show that there exist convex combination of $x_{n}$ that converge to $x$ in $X$ uniformly.
It seems to be closed related to the following theorem:
Suppose $X$ is a metrizable locally convex space. If $\{x_{n}\}$ is a sequence in $X$ that converges weakly to some $x\in X$, then $\exists$ a convex combination of $x_{n}$ that converges to $x$ originally.