I'm still wrapping my head around exactly what 'constructive' mathematics is. To my understanding, there are several theorems in real analysis which depend on the axiom of either dependent or countable choice. These are variants of the axiom of choice, and it was shown my Paul Cohen that neither of these axioms are provable in ZF without the axiom of choice. It follows that they have some nonconstructive affiliation, yet, as they are so much weaker than the full axiom of choice, they do not imply the law of excluded middle, and consequently their use in constructive mathematics is not so widely disputed, but what does this say about those methods or theorems in real analysis which do not use these axioms? Is it true that real analysis is "not entirely nonconstructive", and that only certain parts of it are?
Any clarification and responses are appreciated.