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I am looking for different proofs of the theorem :

If $f$ is a continuous real-valued function on $[a,b]$ and if any $epsilon>0$ is given, then there exists a polynomial $p$ on $[a,b]$ such that

$|f(x)-P(x)|< epsilon $
for all $x in [a,b]$.

My favorite proof uses probability! Here are two exercises that will help you prove it.

Step 1: Let $Z_n$ be a sequence of random variables and $c$ a constant such that for each $epsilon > 0$ it holds
that $$mathbb{P}[|Z_n −c| > epsilon] rightarrow 0, text{as } n rightarrow 0.$$ Show that for any bounded continuous function $g,$
$$mathbb{E}[g(Z_n)] rightarrow g(c) text{as } n rightarrow 0.$$

A hint for Step $1$ is to use the fact that $g$ is bounded and the definition of continuity.

Step 2: Let $f(x)$ be a continuous function in $[0,1]$. Consider the $textit{Bernstein Polynomials,}$ defined by
$$ B_n(x) = sum_{k=0}^n f left( frac{k}n right) dbinom{n}k x^k (1-x)^{n-k}. $$
Show that $B_n rightarrow f$ uniformly.

A hint for Step $2$ is to use the weak law of large numbers.