Prove for all natural numbers a and b, there exists a natural number s such that a < sb (Archimedean Principle)

Prove for all natural numbers a and b, there exists a natural number s such that a < sb.

This is a proof of the Archimedean Principle which states that with a lever large enough a man could move the earth. In other words, with a multiple large enough, a can be surpassed by b.

This is a proof by induction, so we prove the first case which is 1 for the natural numbers.

If b = 1 then we choose s to be a+1. Then a < a+1 = sb, so the first case works.

Now let us assume the general case where b is equal to some natural number n. Then there exists an s such that a < sn < s(n+1), so the statement is true when b=n+1.

Thus, by induction there exists a natural number s such that a < sb for all natural numbers a and b. ~~~~