Consider the following well-known result: Let $(E,\leq)$ be an ordered set. Then the following are equivalent: (i) Every nonempty subset of $E$ has a maximal element. (ii) Every increasing sequence in $E$ is stationary.
It is immediate that (i) implies (ii). To prove the converse, one assumes that (i) is false and then "construct constructs step by step" a strictly increasing sequence.
The common mistake (which I have seen in textbooks) is to describe the latter construction as a proof by induction. In fact, the construction uses the axiom of choice (or at least the dependent choice axiom).
(As a special case, I don't think ZF can prove that every PID is a UFD.)
