Weierstrass Approximation Theorem/Lemma 2

Theorem
Let $\map {p_{n, k} } t : \N^2 \times \closedint 0 1 \to \R$ be a real valued function defined as:


 * $\map {p_{n, k} } t := \dbinom n k t^k \paren {1 - t}^{n - k}$

where:
 * $n, k \in \N$
 * $t \in \closedint 0 1$
 * $\dbinom n k$ denotes a binomial coefficient.

Then:
 * $\ds \sum_{k \mathop = 0}^n \paren {k - n t}^2 \map {p_{n, k} } t = n t \paren {1 - t}$

Proof
From the binomial theorem:


 * $\ds 1 = \sum_{k \mathop = 0}^n \binom n k t^k \paren {1 - t}^{n - k}$

From Lemma 1:

Then: