We know that some functions don't converge but that's only because we can't try n infinite numbers.
That's just patently false. For example, we know that this sequence doesn't converge:
1,
1 + 1/2,
1 + 1/2, + 1/3,
1 + 1/2, + 1/3 + 1/4,
1 + 1/2, + 1/3 + 1/4 + 1/5,
... etc.
We know that won't converge, and not being able to try infinite numbers has nothing to do with it.
So what you say seems to me to be, on the surface, pretty much complete nonsense. To one who is trained in mathematics, what you're saying feels very similar to people who haven't travelled much saying that the Earth is obviously flat.
With respect, this response seems needlessly harsh and supercilious. That’s the kind of stuff that discourages people from going deeper into a field. The question appears to have been asked in good faith ... why not be more charitable and constructive?
And as you probably know, it was discovered that many sequences that supposedly “won’t converge” indeed do converge under analytic continuation, there are reasonable and terribly subtle philosophical questions to ask about the validity of inductive reasoning, and there are working mathematicians who consider themselves finitists.
I mean, you could have your own axiomatic set-theory where mathematical induction doesn't work, but I'm not sure how to construct it without it being a very strange set-theory indeed.
That's not to say that the real world physics will continue to match our models for big n, or anything like that. Just that our abstract mathematical models continue to "work".
Most of the math that we use is self-consistent, and our proof system is, as far as we can tell, relatively consistent with the known properties of the universe.
What I'm getting at is that no, we don't actually KNOW with 100% certainty that, for example, certain proven divergent series continue to diverge at infinity, if your standard for proof requires trial of an infinite span of numbers; but such speculation is so far out of the realm of what is known and "proven" according to our [mostly] self and externally consistent set of knowledge[1] that it is more or less in the realm of pseudoscience/metaphysics.
1. Excluding the Incompleteness Theorem
We know that some functions don't converge but that's only because we can't try n infinite numbers. How are we certain that these non-convergent functions do not converge to some value and that math breaks at big scales (or smaller scales). =much like our universe=