My question comes from the necessity of proving that one of the hypothesis in the Arzela Ascoli theorem fails in the example:
$f_n(x)=\frac{1}{1+(x-n)^2}; x\in [0,\infty)$
I have tried proving it is not uniformly equicontinuous nor uniformly bounded but keep failing, so my only assumption is that Arzela Ascoli cannot be applied because $[0,\infty)$ is closed but not bounded, so it is not compact in $\mathbb{R}$. Yet, I cannot think of a way to show the theorem fails in this example.
Best Answer
This sequence is equi-continuous and uniformly bounded. The only reason Arzela-Ascoli Theorem is not applicable here is because $[0,\infty)$ is not compact.
Note that $0 \leq f_n(x) \leq 1$. So the sequence is uniformly bounded. Let su prove equi-continuity: $|f_n(x)-f_n(y)| \leq \frac {|x-y| (|(x-n)+(y-n)|} {(1+(x-n)^{2}) (1+(y-n)^{2})}$. Use the inequality $|a| \leq \frac 1 2 (1+a^{2})$ with $a=x-n$ and $a=y-n$ to see that the sequence is equi-continuous.
Note that $f_n(x) \to 0$ for each $n$ whereas $f_n(n)=1$. This shows that the sequence has no uniformly convergent subsequence.