I want to prove this:
If $f$ is a continuous function on $[0,1]$ then there exists a sequence of polynomials $f_n$ converging to $f$ uniformly.
My notes tell me to set $(X_i)_{i \ge 1}$ iid $\mathrm{Ber}(p)$ and set $\displaystyle f_n(p) = \mathbb E_p [f(S_n/n)] = \sum_{k = 0}^n \binom n k p^k (1 - p)^{n - k} f(k/n)$ and then use Chebyshev's inequality to establish uniform convergence. (of course with $\displaystyle S_n = \sum_{i = 1}^n X_i$)
This is only shortly after covering the weak law of large numbers and probabilitistic convergence, nothing else covered at this point. I know that Chebyschev's inequality concerns probabilities of the form $\mathbb P(|X - \mathbb E[X]| < k)$, but I'm not sure what to pick $X$ to be. I'm assuming we're using this inequality to establish some sort of probabilistic convergence relevant to $f$ being uniform. $X = f(S_n/n)$ doesn't seem particularly useful since then we have two things dependent on $n$.
Any ideas on where to start?
