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Message: [QUOTE="Jacob Fugal, post: 4525304"] And here's the "below" referred to (I forgot it earlier). :) Remember that i >= 4. Let's take i = 4 as a basis case. 81 * 4 = 324 < 1000 = 10^(4-1) Now, assume the hypothesis for some i >= 4. We will prove the hypothesis continues to hold for j = i + 1. Since 1/9 < i; 81 < 9 * 81 * i. Adding 81 * i to both sides of that inequality we get: 81 * j < 10 * 81 * i From the other side, we can start with the inequality for i (81 * i < 10^(i-1)) and multiply both sides by 10 to get: 10 * 81 * i < 10^(j-1) Combining those two inequalities, we have: 81 * j < 10^(j-1). Isn't induction great? :) Jacob Fugal [/QUOTE]