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
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 $\epsilon.$