Let $P(n)\,$ be a statement concerning the positive integer $n\,$. Suppose that:

(i) (Basis of induction) $P(1)\,$ is true;

(ii) (Induction step) If $P(n)\,$ is true for some $n\,$, then $P(n+1)\,$ is also true.

Then $P(n)\,$ is true for all positive integers $n\,$.

Remarks. (i) The basis of induction checks that we can start the argument with a true statement; and

(ii) The induction step asserts that, having reached some point, we can always take one further step ahead.