Consider the function sin($\displaystyle \pi$$\displaystyle x$) on [-1,1] and its approximations by interpolating polynomials. For integer $\displaystyle n$$\displaystyle \geq$1, let $\displaystyle x_{n,j}=-1+\frac{2j}{n}$ for $\displaystyle j=0,1,...,n$, and let $\displaystyle p_{n}(x)$ be the $\displaystyle n$th-degree polynomial interpolating sin($\displaystyle \pi$$\displaystyle x$) at the nodes $\displaystyle x_{n,0},...,x_{n,n}$. Prove that

$\displaystyle \max_{x\in[-1,1]}\left | \textup{sin}{(\pi{x})-p_{n}{(x)}} \right | \to 0$ as $\displaystyle n \to \infty$