Relation of Lagrange Interpolation to Windowed Sinc Interpolation

For an infinite number of equally spaced samples, with spacing $x_{k+1}-x_k = \Delta$, the Lagrange-interpolation basis polynomials converge to shifts of the sinc function, i.e.,

$$l_k(x) =$$   sinc$$\left(x-k\Delta\over\Delta\right), \quad k=\ldots,-2,-1,0,1,2,\ldots$$

Proof: As order $\to\infty$, the binomial window $\to$ Gaussian window $\to$ constant (unity).

Alternate Proof: Every analytic function is determined by its zeros and its value at one nonzero point. Since $\sin(\pi x)$ is zero on all the integers except 0, and since sinc$(0)=1$, it therefore coincides with the Lagrangian basis polynomial for $N=\infty$ and $k=0$.