Uniform Continuity and partial sums equation proof

Question

Given $f$, a uniformly continuous function defined on the interval $[0,1]$, I need to prove that $$\lim_{n\rightarrow \infty} \frac{1}{2^n} \sum_{k=1}^n (-1)^k \binom{n}{k} f(k/n)=0.$$ I have tried tackling this exercise from a couple of angles but I seem to lack the intuition and technical skills to crack this egg open, so I am at your mercy.

Answer 1

1. As $f$ can be expressed as a uniform limit of polynomials, by linearity it's enough to show the result when $f(x)=x^p$ where $p$ is a non-negative integer.

2. We can show this by induction on $p$, using the relationship $$\frac{k^p}{n^p}\binom nk=\frac{k^{p-1}}{n^{p-1}}\binom{n-1}{k-1}.$$