Diference between the Real numbers and a set

Question

I was doing a exercise from my calculus class that aks for me to prove that a series converge uniformly in the set $[-r,r]$ for all $r>0$. My question is, what is the difference between set $[-r,r]$ for all $r>0$ and the set of real numbers? Naively, it look like for me that those two should be the same.

Answer 1

Uniform convergence is a property of not just the function, but the space it is operating on: you need to be able to control the speed of convergence simultaneously at every point in the space.

Whenever you expand the set you are talking about, it can only make uniform convergence more difficult: maybe a specific $\delta$ worked when you were only considering $[-1,1]$, but the function grows more quickly outside of that interval than inside it.

So, it is entirely plausible to say that you have uniform convergence on any given closed interval, no matter how big, but that you don't have uniform convergence on $\mathbb{R}$.

For an example, consider the sequence $$ f_n(x):=\frac{e^{\lvert x\rvert}}{n}. $$ Clearly, this converges pointwise to the constant $0$-function as $n\to\infty$. The convergence is uniform on any fixed interval $[-r,r]$, because $$ \lvert f_n(x)-0\rvert\leq\frac{e^r}{n}\to0 $$ independent of the specific $x\in[-r,r]$ chosen.

However, this does not converge uniformly on $\mathbb{R}$: no matter what $\epsilon$ you choose, no matter how large you let $n$ be, you can always choose $x$ sufficiently large to make $\frac{e^{\lvert x\rvert}}{n}>\epsilon$.