Hi!

Problem:Show that $\displaystyle f_{n}(x)=nx^{n}(1-x) $ converges "pointwise", donīt know if this is the correct english word for it. And show that the convergence is not uniform on $\displaystyle [0,1]$ .

Solution:

Itīs easy to find that $\displaystyle f_{n}(x) \to f(x)=0 \; \; , \forall x \in [0,1] $

How do I show that the convergence is not uniform?

We have that $\displaystyle \lim_{n\to \infty} f_{n}(x) = f(x) $ is continous on $\displaystyle [0,1]$ , so we have to proceed right?

By letting $\displaystyle d(x)=|f_{n}(x)-f(x)| $ , we have to show that $\displaystyle d(x) \to 0 $ as $\displaystyle n\to \infty $

I tried taking the derivative but didnīt come up with anything good.

Thx!