I came across this proof in Spivak that appears to use complete (strong) induction. He is trying to argue that a set A of natural numbers with no least member is the empty set. He lets another set B be the set of natural numbers 1, ..., n that are NOT in A. For the base case, 1 is in B since otherwise 1 would be the least member in A. Then he assumes that 1, ..., k are not in A, then k + 1 is not in A (otherwise, k+1 is the least member of A). Hence, the set B contains all the natural numbers, all not in A, so A is the empty set. Is this an example of complete induction? Do you use complete induction when it is advantageous or clear that the numbers between 1 and k are important in understanding the proof? For instance, this proof emphasizes the notion that the numbers between 1 and k (and also 1 and k) could not be in A, so it makes it clear why k + 1 would be the least member if it was in A. Although I guess it wouldn't really be that different if we used regular induction. You would still precede from 1 to 2 to 3 to show that the statement holds for all natural numbers, but does complete induction help to reduce the perceived gap between 1 and k?