I thought many results in calculus need axiom choice. For example, I thought one needs AC to prove that a bounded sequence in the real line has a convergent subsequence. Recently I was taught that one only needs mathematical induction to prove it. So here are my questions.
Can most results in calculus be proved without AC? If yes, what are the exceptions, to name a few?
Obviously a theorem which uses Zorn's lemma most likely does need AC. So please exclude obvious ones.
Edit By "without AC", I mean without any form of AC, i.e. countable or not, dependent or not. In other words, within ZF.
Edit One of the motivations of my question is as follows. People often unconciously use AC to prove theorems. And it often turns out that their uses of AC are unnecessary. For example, an infinite subset of a compact metric space has a limit point. In his book "Principles of mathematical analysis", Rudin proves this by choosing a suitable neighborhood of every point of the space. He uses AC here, though he doesn't say so. However, since a compact metric space is separable, you can avoid AC to prove this theorem.
Edit I'll make the above statement clearer. You can even avoid countable AC to prove the above theorem. In other words, you can prove it within ZF.
Edit I'll make my questions clearer and more specific. By calculus, I mean classical analysis in Euclidean spaces. Take, for example, Rudin's "Principles of mathematical analysis". Can all the results in this book be proved within ZF? If not, what are the exceptions?