Give an example of an assertion which is not true for any positive
integer, yet for which the induction step holds.
First of all, definition.
In inductive step, we suppose that $P(k)$ is true for some positive
integer $k$ and then we prove that $P(k + 1)$ is true.
My thoughts : Since the assertion has to satisfy the induction step, it must be false for $n=1$.
But I cannot think of any such assertion.
Does anyone have any hints? Thank you.
Edit: Just realised that my thoughts don't make much sense, as the question demands the assertion to be false for every positive integer.
Best Answer
There is no shortage of good answers to this question, so I'll give a couple examples in the hopes that thinking about them will help you to produce your own answers.
Two possibilities for $P(n)$ are
-the assertion that $n! = 0$
-the assertion that $n > 2n$