I think the set needs to be bounded below to avoid occurrence of infinite descending chains for induction to be applicable but couldn't decide if it can be bounded above as well.
Inductive step is P(n) implies P(n+1). But if the set on which we are applying induction contains a greatest element, say a. And if P(a) is true then what can we say about P(a+1). I think induction would still be applicable whether or not a+1 belongs to the set.
I would really appreciate if an example could be provided in which induction is being applied on a set bounded above.