Elementary proof about nth differences of nth powers of integer

Question

In a post on Math.SE., a proof sketch was proposed for the proposition below:

The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$.

Anyone familiar with algebraic identities and induction can understand that proof, but it's not the case for a 7th-grade students!

Is there an elementary proof for this proposition?

Thanks.