In this exercise you are to prove some well-known facts about numbers as a way of demons...

In this exercise you are to prove some well-known facts about numbers as a way of demonstrating use of the WOP. Use the WOP to prove the following:

(a) If a > 0, then for every natural number n, aⁿ > 0,

(b) For all positive integers a and b, b ≠ a + b. (Hint: Suppose for some a there is b such that b = a + b. By the WOP, there is a smallest such that, for some b, b = a₀ + b. Apply the WOP again.)

(c) is irrational.