I know the definition of "pointwise convergence" and "uniform convergence", nevertheless I have some difficulties understanding the difference between those two concepts.
My book defines Uniform convergence as follows:
Let be $f_n$ a sequence of functions on $A\subseteq \mathbb{R}$. Then $(f_n)$ converges uniformly on $A$ to a limit function $f$ defined on $A$ if, for every $\epsilon>0$ there exists $N\in\mathbb{N}$ such that $|f_n(x)-f(x)|<\epsilon$ whenever $n\geq N$ and $x\in A$.
Now I do understand that in the case of uniform convergence we can choose $N$ irrespectively of the point $x$ while in the case of Pointwise convergence this is not the case (something like the difference between continuity and uniform continuity).
If the sequence of functions converge pointwise to $f$ it means that $\forall \epsilon>0, x\in A$ there exists $N$ such that $|f_n(x)-f(x)|<\epsilon$ whenever $n\geq N$, right? In this case $N$ depends both on $\epsilon$ and $x$.
My question is, given $\epsilon$ couldn't we pick an $N^*= \sup_{x\in A}{N(x,\epsilon)}$ (maybe this notation is a little bit messy).
Then we know that for this $\epsilon$ $, |f_n(x)-f(x)|<\epsilon$ whenever $n\geq N^* \forall x\in A$.
I know that in some cases this cannot be done, but I cannot see whether this is always false.
Would you mind to spot my error here? Thanks in advance.
The value $N^*$ might not exist in $\mathbb{N}$ (i.e. it's $\infty$). You write $\max$ but really you should only write $\max$ when the underlying set (in this case, $A$) is finite. If it is not you must write $\sup$ and the $\sup$ is not always attained. If the $\sup$ is attained, i.e. $N^*(\epsilon)$ is finite, for all $\epsilon$ then there is no issue with choosing $N^*$ and you have uniform convergence.