Let $\Delta x_ {k} = x_{k} - x_{k+1} $ and the convention $\Delta ^{1} = \Delta$ where $x_ {k}$ is a sequence.
So $\Delta^{2} x_ {k} = \Delta ( \Delta x_{k} ) = \Delta (x_{k} -x_{k+1}) $ $= x_{k} - x_{k+1} -(x_{k+1} -x_{k+2}) = x_{k} -2x_{k+1} + x_{k+2}$.
Argue that the $\Delta$ function is linear for addition and prove
$\Delta^{k} a_{m} = \sum_{j =0}^k (-1)^{j} \binom{k}{j} x_{j} $ where $a_{m}$ is the arithmetic mean.
I guess the best and easiest way to prove this is by induction but I can't see how to do it.
The theorem is from this paper where the argument is too unclear for me. Page 20/63 http://www.cs.umb.edu/~offner/files/pi.pdf