Assume $f_n(x)=x^n$
Let $g(x)$ be continuous on $[0,1]$ with $g(1)=0$
Prove ${g(x)f_n(x)}$ is uniformly convergent on $[0,1]$
I know how to prove this using Dini's theorem about compact sets and decreasing functions. But I want to know if there's another way to prove it.