I have a final exam in Analysis approaching and one of the question is that the professor will give us a sequence of functions and we will have to prove that they converge point-wise or uniformly by definition ($\forall \epsilon$...)

My question is that when it comes to writing the point-wise or uniform convergence proof is the process the same for both? Meaning, after choosing a N, the only difference is that the structure of their proofs are different;

Pointwise: Let $\epsilon > 0$ and let x $\in$ A. Choose N = ...

Uniform: Let $\epsilon > 0$. Choose N = ...

is this the only difference

  • $\begingroup$ This is not so different that asking if the only differences between $\pi$ and $e$ are that the shapes with which they are written are different... Uniform and pointwise convergence are two different concepts, of course the way they are expressed are different... Anyway, if you just want to know the how-to-do-it then yes, for the pointwise convergence you are allowed to choose an $N$ which depends of $x$. For the uniform convergence $N$ must not depend of $x$, i.e. it must be uniform in $A$. $\endgroup$
    – user378947
    Dec 18 '16 at 4:59

You should start the proof by taking any $x$ in the set and conclude the proof by saying "so this is true for every point $x.$"

Pointwise. Here $N$ depends both in your chosen point $x$ as well as in $\epsilon.$ i.e., $N=N(x,\epsilon).$

Uniform. In contrast, here your $N$ depends on only $\epsilon.$


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