How can I prove the sequence of functions $x(1-x), x^2(1-x), ...$ converges uniformly on $[0,1]$?

I know from my earlier question [here](https://math.stackexchange.com/questions/549842/prove-the-sequence-of-functions/549863#549863) that I can prove it by taking the derivative. But how can I prove it without taking the derivative since we still haven't defined "derivative" yet in my class? I know I can divide the regions into two pieces.