# Is this statement true or false?

• Apr 22nd 2010, 07:21 AM
paupsers
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 $\{f_n\}$ on the interval [0,1] by $f_n(x)=x^ng(x)$. Does it follow that the sequence of functions $\{f_n\}$ 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?
• Apr 23rd 2010, 11:55 PM
Failure
Quote:

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 $\{f_n\}$ on the interval [0,1] by $f_n(x)=x^ng(x)$. Does it follow that the sequence of functions $\{f_n\}$ 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 $g(x) := x$, for example: because if $f_n$ were to converge uniformly, the limit function would have to be continuous, but it isn't continuous at $x=1$.