Weierstrass Approximation Theorem/Proof 2

Proof
, assume $\Bbb I = \closedint 0 1$

For each $n \in \N$, let:
 * $\ds \map {P_n} x := \sum_{k \mathop = 0}^n \map f {\dfrac k n } \dbinom n k x^k \paren {1 - x}^{n - k}$

We shall show that $\lim_{n \to \infty} \norm { P_n - f}_\infty = 0$.

Let $\epsilon \in \R_{>0}$.

By Heine-Cantor Theorem, there is a $\delta \in \R_{>0}$ such that:
 * $\forall x,y \in \Bbb I : \size {x - y} \le \delta \implies \size {\map f x - \map f y} \le \epsilon $

Let $p \in \Bbb I$.

Let $Z_n$ be a random variable such that:
 * $\ds n Z_n \sim \Binomial n p$

where $\Binomial n p$ denotes the binomial distribution with parameters $n$ and $p$.

Observe that:

Furthermore:

On the other hand:

Therefore:

Thus for all $n \in \N_{> 2 \delta^2 / \norm f_\infty}$ we have:
 * $\size {\map {P_n} p - \map f p} \le 2 \epsilon$

As the above is true for all $p \in \Bbb I$, we have:
 * $\forall n \in \N_{> 2 \delta^2 / \norm f_\infty} : \norm { P_n - f}_\infty \le 2 \epsilon$