Writing and checking a complete proof

Begin by naming the objects and their domains, then distinguish hypotheses from the desired conclusion. Quantifiers determine which objects are arbitrary and which may be chosen as witnesses. If a theorem says every nonzero real x has a property, the proof must not begin by choosing x positive without also treating negative x. If it says some object exists, an arbitrary example is useful only after all required properties have been checked. A correct calculation can still address the wrong logical claim.

Choose a proof structure that matches the task. A conjunction needs both parts; an equivalence needs both implications; an existence-and-uniqueness theorem needs a witness and a proof that no different witness works. A set equality can be proved by two inclusions. Induction needs initialization and a uniform preservation step. Name the structure early enough that the reader understands why each paragraph appears and can see when the proof’s obligations have all been discharged.

Not every algebraic move preserves exactly the same solution set. Dividing by x assumes x is nonzero. Squaring an equality forgets a sign condition. Taking a square root requires a nonnegative argument in real arithmetic. An inequality multiplied by a negative quantity reverses direction. During an audit, stop at each such operation and state its validity conditions. If a condition is unavailable, split cases or treat the result as a candidate that must be checked in the original statement.

Likewise, cite the actual premise used by a logical inference. A universal theorem can be instantiated only at an object in its domain. An induction hypothesis can be applied only to indices already covered. An existential witness cannot silently be reused as if it worked for every possible input. A line-by-line proof check asks both “does this follow?” and “from which established facts?” Marking dependencies often reveals that a promising-looking argument assumes exactly the conclusion it set out to prove.

Small cases, boundary values, and numerical plots are effective debugging tools. Test zero, negative values, equality cases, empty collections, and the first admissible induction index when those objects belong to the domain. A single exact counterexample can refute a universal claim, while many successful tests ordinarily cannot prove one over an infinite domain. Floating-point output adds another limitation: an apparently zero residual can be roundoff, not an exact identity. Use exact arithmetic when the alleged counterexample depends on equality.

End by connecting the final established fact to the original goal and discharging temporary assumptions. Ask a reader to restate the argument’s central reason in one sentence; if only a sequence of unexplained formulas remains, add a bridge rather than extra decoration. Finally distinguish proving a theorem within a model from establishing that the model describes an experiment. A flawless queueing calculation still relies on its arrival and service assumptions. Good proof writing identifies the guarantee’s boundary as carefully as its conclusion.