Uniform convergence of $x^n-x^{2n}$

Question

Does the following sequence of functions converge uniformly when $x$ is in $[0,1]$:

$$f_n(x)=x^n-x^{2n}$$

I know the answer is that it doesn't, but I couldn't figure out why. Would be glad to hear an explanation. Thanks!

Answer 1

The sequence converges pointwise to the function which is $0$ for all $x\in[0,1]$.

In order to show that convergence is not uniform, you should check that $$\sup_{x\in[0,1]}|(x^n-x^{2n})-0|$$ does not tend to $0$ as $n\to\infty$. This is continuous and $[0,1]$ compact, so the $\sup$ is actually a $\max$.

For fixed $n$ you can use calculus (or AM-GM, if you know it) to find the maximum value of $x^n-x^{2n}$ on $[0,1]$. (You should find this is $1/4$ for every $n\geq 1$, which certainly does not tend to $0$.)