# Thread: Is this statement true or false?

1. ## Is this statement true or false?

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?

2. Originally Posted by paupsers
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?
The conclusion fails for $\displaystyle g(x) := x$, for example: because if $\displaystyle f_n$ were to converge uniformly, the limit function would have to be continuous, but it isn't continuous at $\displaystyle x=1$.