Define the functions $f_n: [0,1] \to \Bbb R$ by $f_n(x)=n^px\exp(-n^qx)$, where $p,q>0$.
Show $f_n\to 0$ pointwise. Find $||f_n||_{\infty}$.
So, I have shown that it converges pointwise to $0$ and that $||f_n||_{\infty}=\frac{n^{p-q}}{e}$
So for $p<q$ $f_n\to 0$ uniformly.
I am asked whether it converges uniformly on the interval $[0,0.99]$ or on the interval $[0.01,1]$.
Well, my answer is yes, why wouldn't it? I don't understand why it wouldn't? Anything to help me understand this would be of big appreciation!
You may be making an error in your thinking. On the interval $[.01,1],$ we have
$$0\le f_n(x) \le n^p\cdot 1 \cdot \exp (-n^q\cdot .01).$$
The right side $\to 0.$ Thus $f_n\to 0$ uniformly on the interval $[.01,1],$ no matter what $p,q$ are.
Now for uniform convergence on all of $[0,1],$ you found that this happens iff $p<q.$ But we just found that on $[.01,1],$ we have uniform convergence no matter what $p,q$ are. The point of this problem is to get you thinking about this.