## interpolating polynomial for sin(pi*x)

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

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