One could use a technique similar to induction to prove a proposition about all integers by showing that S(0) is true, and then showing that:

S(n+1) is true whenever S(n) is true.

S(n-1) is true whenever S(n) is.

From this we can conclude that S(n) is true for all integers n.

(All we are really doing is applying induction twice separately.)

In fact induction as you know it can be adapted to any set of the same "size" as the positive integers. You have probably already seen this when you have proven things about all EVEN integers say.