Let $ \{f_n\}$ be a sequence in $C(K)$ where $K$ is a compact Hausdorff space with $|f_n| \le 1$ for $n=1,2,......$ . If $f \in C(K)$ and $ f_n \to f$ pointwise on $K$ then show that there exist some sequence of convex combinations of the $f_n$ 's converge uniformly to $f$.
I know that $C(K)$ in complete under sup norm.But I am unable solve this using the completeness..Need some help.
This is an overkill I guess. Let $\mu$ be a Radon measure on $X$, then $\mu(K) <\infty$. Also $\| f_n\|_{L^1}$ is uniformly bounded by $\mu(K)$ as $|f_n|\le 1$. By Lebesgue's dominated convergence theorem, $$\tag{1} \int_K f_n d\mu \to \int_X f d\mu.$$ By Riesz' representation theorem, all bounded linear functional on $C(K)$ is given by Radon measure, thus $(1)$ implies that $\{f_n\}$ converges weakly to $f$. Now the claimed result follows from Mazur's lemma, which claimed that if $\{f_n\}$ is a sequence in a Banach space $X$ which converges weakly in $X$ to $f$, then a convex combination of $\{f_n\}$ converges strongly to $f$ in $X$. Now our Banach space is $X = (C(K), \| \cdot \|_\infty)$, so the conclusion is that a convex combination of $\{f_n\}$ converges uniformly to $f$.