How do I decide between direct, contrapositive, and contradiction proofs?

[Lily.D]

I'm a second-year math major currently taking my first rigorous course in real analysis, and I'm struggling to internalize the different proof techniques, particularly when to apply a direct proof versus a proof by contradiction or contrapositive. I understand the definitions in isolation, but when faced with a statement like "if a sequence is Cauchy, then it is bounded," I freeze up and don't know how to start structuring my argument. For students who have successfully made this transition to higher-level mathematics, what mental frameworks or practice strategies helped you most? How do you systematically break down a proposition to decide on an approach, and are there any specific resources or problem sets that helped build your intuition for constructing clean, logically sound proofs?

[Jonathan99]

Great topic. A simple mental model: for a statement like 'If P then Q', try the contrapositive first; if that feels awkward, build a direct proof and only then turn to a contradiction. For the classic fact 'Cauchy implies bounded', the clean path is direct: use the Cauchy condition to bound the tail, then bound the finite head.