Something about Apologetics.

[Categories+Sheaves]

In fact, if you are educated in calculus you should know that limits can be said to 'exist' even when they don't. In other words, if a function has a missing point (an undefined point) but it is continuous on either side of that point up to that point, then that point can be proven to be a 'limit'. Yet clearly that does not mean that the point must 'exist'.

Absolutely. Not all spaces are sequentially compact, and not all metric spaces are complete. But this observation tout court has nothing to do with the validity of a notion of a space with these properties.

Zeno's claim is that you can't perform an infinite number of finite steps. And the calculus limit doesn't challenge that, nor does it show that Zeno was wrong.

How does a methodology for describing infinite sequences/sums/etc. not challenge the claim that you can't perform an infinite number of finite steps?

To begin with Zeno was simply arguing that IF the world is a continuum, then motion would not be possible.

Which is funny, because the move to a continuum (as Zeno's paradoxes don't rely on any numbers outside the rationals) is precisely what you need to resolve the paradoxes.