I've got a confusing problem about consistent convergence in my homework:
Study the consistent convergence of the following columns of functions on the specified interval: $f_n(x)=e^{-(x-n)^2}, I_1 = (-l, l)(l \in R^+), I_2 = (-\infty, \infty)$. It's clear that $\lim_{n\rightarrow \infty}f_n(x)=f(x)\equiv0$, thus $f_n(x)$ converges uniformly on $I_1$ while it doesn't remains when it comes to $I_2$.
In my opinion, this leads to a paradox that if an assignment is true for any centrosymmetric interval, then it must be true for the whole Number Axis, on the contrary to what this example shows. Are there any errors in my calculations? Or is this extension not absolutely correct?Looking forward to anyone coming to help.
This problem hit me when I was reading a proof of why the function $e^x$ has a Maclaurin Series. The author made the prove on the interval $(−R,R)$ first, as $R$ is a positive number, then made a extension to the whole number axis. What's the key reason that the same approach works for this question above? Are there any discriminatory methods that can help me determine the scope of this extension?