Mazur's Lemma (can't find a proof anywhere)

If we have a normed vector space $X$, and $x_n \to x$ weakly, then it's clear how to show there's a sequence $y_n \to x$ strongly (since the weak closure and strong closure of a convex set coincide, just consider the closure of the convex hull of the sequence).

I'd like to know how to prove the second half of the lemma i.e. you can choose the $y_n$ to lie inside the convex hull of $\lbrace x_1,x_2,..,x_n \rbrace$. Why is this?

Solution 1:

The way I would do it is to notice that $$ \text{conv}\{x_n\}_{n\in\mathbb{N}}=\bigcup_{k=1}^\infty\text{conv}\{x_1,\ldots,x_k\}, $$ and then of course the closures are equal. Since $$ x\in\overline{\bigcup_{k=1}^\infty\text{conv}\{x_1,\ldots,x_k\}}, $$ we have $$0=\lim_{n\to\infty}\text{dist}(x,\bigcup_{k=1}^n\text{conv}\{x_1,\ldots,x_k\})=\lim_{n\to\infty}\text{dist}(x,\text{conv}\{x_1,\ldots,x_n\}) $$(the numbers inside the limit form a decreasing sequence of nonnegative numbers with inferior limit equal to zero). So we can choose a sequence $\{y_n\}$ with $y_n\in\text{conv}\{x_1,\ldots,x_n\}$ and $\lim_{n\to\infty}\|x-y_n\|=0$.