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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 $x \in [a,b]$.

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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.

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  • $\begingroup$ I have seen the proof which uses only Bernstein Polynomials . I am looking for a proof that uses Lagrange Interpolating Polynomial . Thanks for your post , I will try to prove it. $\endgroup$ – mike moke Dec 2 '18 at 6:22

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