Prove each of the following for every positive integer n. Use steps (a)–(e) as in Exercises.

If 0 < a < 1, then aⁿ < a^n-1.

Steps

Let Sn represent the given statement, and use mathematical induction to prove that Sn is true for every positive integer n. Follow these steps.

(a) Verify S₁.

(b) Write S_k .

(c) Write S_k+1.

(d) Assume that Sk is true and use algebra to change S_k to S_k+1.

(e) Write a conclusion based on Steps (a) – (d).