I'm trying understand the proof of the Arzela Ascoli theorem by this lecture notes, but I'm confuse about the step II of the proof, because the author said that this is a standard argument, but the diagonal argument that I know is the Cantor's diagonal argument, which is used in this lecture notes in order to prove that $(0,1)$ is uncountable and this is an argument by contradiction while the diagonal argument of the proof of the Arzela Ascoli theorem is a direct argument.
My doubt is in what sense the argument of the step II is a 'standard diagonal argument'? Is this argument in the sense that the diagonal is used in order to construct an element which has the desired property? It seems make sense to me because the diagonal in the proof that $(0,1)$ is uncountable is used to construct an element (a number) in $(0,1)$ which has the property that this number is not on the list of the numbers in $(0,1)$ while the diagonal used on step II of the proof of the Arzela Ascoli theorem is used to construct an element (a sequence) which has the property that converges pointwise to the dense subset $S$ constructed on step I of the proof.
Other thing that I thought about the relation between the standard diagonal argument and the Cantor's diagonal argument is just because we construct a tabular form in both cases, but the connection between these two arguments is not so close to the point of being relevant as the author of the answer in this topic comments.
Thanks in advance!
$\textbf{P.S.:}$ I don't know if I'm being clear enough, but my doubt is not about the use of diagonal argument in the step II of the proof of the Arzela Ascoli theorem, but why the argument receives this name since the Cantor's diagonal argument it's an argument by contradiction and the argument in the step II of the proof is a direct argument.