Rigorous writing and classic pitfalls

Classic mistakes to avoid

Mistake number 1: forgetting the base case

The most frequent mistake is to prove only the inductive step, thinking that this is enough. But a perfectly correct inductive step, without a checked base case, proves nothing.

Example of an inductive step without a base case (classic pitfall):

Let P(n) : "n = n + 1" (a property obviously false for every n)

Inductive step: if we assume P(k) true, that is k = k+1,
           then adding 1 to both sides: k+1 = k+2,
           which is exactly P(k+1).

(the inductive step is "proved" while P(n) is false for every n: without a base case, the inductive step alone is worthless)

No domino falls if nobody pushes the first one: without an explicitly checked starting rank, the chain of implications never begins.

Mistake number 2: an inductive step that does not rely on the hypothesis

If the proof of P(k+1) never brings in the hypothesis P(k), it is often the sign that P(k+1) has been re-proved directly (which is not proof by induction) or that there is a hidden logical error.

Mistake number 3: wrong starting rank

If the property is only true from a rank n0 different from 0 (for example n0 = 3), you must initialise precisely at that rank n0, and not at 0 or 1. Checking P(0) when P(0) is false (but P(3) is true) would completely invalidate the proof.

Mistake number 4: confusing "assume P(k)" and "prove P(k) for all k"

The induction hypothesis is about ONE fixed, arbitrary rank k, not about all ranks at once. You never assume the property is true everywhere from the start: that would be assuming exactly what you are trying to prove, a logical error known as circular reasoning.

Common pitfall (summary)