Why doesn’t diagonalization requiring taking a limit?

When we quantify infinite sums, we do so by taking the limit as $ i$ goes to infinity. For example, we look at $ \lim_{i\rightarrow \infty}\sum_{i\in \mathbb{N}}i$ , and then we say that this diverges.

When we do diagonalization, we iterate over an infinite list while indexing each list item by a natural number, and then talk about the result. Why can we do this without invoking limits? Shouldn’t we speak of the result of the diagonalization as a limit?