proof

A proof is like an unbroken chain of stepping stones from the near bank (what we already accept) to the far bank (the claim). Each stone must actually bear weight, and each must be within a single sure step of the previous one — no leaps of faith across open water.

Formally, a proof is a deductive argument that establishes a statement with certainty: a finite sequence of assertions, each of which is an axiom, a definition, a previously proved theorem, or follows from earlier assertions by a valid rule of inference. Once a theorem is proved from accepted axioms, it is true wherever those axioms hold, with no exceptions and no probabilities.

Analysis relies on a handful of recurring proof strategies: direct deduction, proof by contradiction, the contrapositive, induction, and the epsilon–delta template. A correct proof is not the same as a convincing story — checking validity, not plausibility, is the discipline. The honest caveat: a proof guarantees the conclusion follows from the hypotheses, but it can hide an error in a single mis-stated lemma, which is why hypotheses must be tracked with care.