The definition of pointwise convergence of a sequence of functions $\{f_n\}$ on a set $A\subseteq\mathbb{R}$ to a funtion $f$ is that $\forall x\in A$ the sequence $\{f_n(x)\}_{n\in\mathbb{N}}$ converges to $f(x)$. That is, $$\forall x\!\in\!A\ \ \forall\epsilon\!>\!0\ \ \exists N\!\in\!\mathbb{N}\ \ \forall n\!\geq\!N\ \ |f_n(x)-f(x)|\!<\!\epsilon$$ My question is: Isn't it same as the one given below? $$\forall\epsilon\!>\!0\ \ \forall x\!\in\!A\ \ \exists N\!\in\!\mathbb{N}\ \ \forall n\!\geq\!N\ \ |f_n(x)-f(x)|\!<\!\epsilon$$ I think we should be able to change the order of two nearby universal quantifiers since the variables cannot depend on one another. But my guide says the order of $\forall x\!\in\!A$ and $\forall\epsilon\!>\!0$ matters here, and he mentioned that there is a counterexample. Am I missing something here?
Note1: We both knew that uniform convergence is different from point-wise convergence. He maintained that the above two are indeed different.
Note2: $\forall x\forall y$ is different from $\forall y\forall x$ if the domain for one of the variables is defined in terms of the other (eg: In the case of $\forall x\!\in\!\mathbb{N}\ \forall y\!\in\!\mathbb{N}\!-\!\{x\}$, it is evident that we cannot change the order). But except for this, I don't know any other problem that makes the two orders different. For, the two statements given above in equation environments, there is no issue of domain of one variable depending on the other. Is there any other problem?