Suppose that g is a continuous function on the interval [0,1] with g(0)=0. Define a sequence of functions $\displaystyle \{f_n\}$ on the interval [0,1] by $\displaystyle f_n(x)=x^ng(x)$. Does it follow that the sequence of functions $\displaystyle \{f_n\}$ converges uniformly on [0,1]? Either prove this is true or give an example of a function g for which the conclusion fails.

Any help on this?