I am not sure how to prove complete induction:
P(n) is a statement of variables n. If
(i) P(1), P(2),...,P(m) are true and
(ii) if P(i) is true then P(k+1) is true.
How do I prove this?
From what I've seen you prove it from having already established weak induction. That looks like the route taken in emakarov's link. Do you already have weak induction for this proof? If not, prove weak induction, and then prove strong induction. Does anyone know of a more direct proof (i.e., prove strong induction without weak induction)?