I am new to this topic(sequence and series of functions) so I have a doubt.All theorems related to uniform convergence which I am studying are given on closed interval for example
(1) If a sequence of functions $f_n(x)$ converges pointwise to a function $f$ on $[a,b]$ then $$Sup_{[a,b]}\vert f_n(x)-f(x)\vert= 0\Rightarrow f_n(x) \text{converges uniformly on [a,b]}$$
(2) Dini's theorem: For an increasing sequence $F={f_n}$ of continuous functions on an interval $I=[a,b]$ which converges pointwise on $I$ to a continuous function $f$ on $I$, Dini's theorem states that $F$ converges to $f$ uniformly on $I$
My questions:(1)Is the study of of uniform convergence not important for unbounded intervals?if it is not so then why we are only studying the convergence on bounded intervals(in particular closed intervals).
2)Are the methods to test the uniform convergence remains same if we work on unbounded intervals,if it is not so then what is the best method to check uniform convergence on unbounded intervals?
Any help would be appreciable
Thanks in advance
Best Answer
There is a lot of difference between bounded and unbounded domains. Dini's Theorem is stated for compact intervals because it is not valid for unbounded intervals. A very simple example to illustrate the difference between the two casees is $f_n(x)=\frac x n$. This sequence converges to $0$ uniformly on any bounded interval but not on $\mathbb R$.