Math 104: Introduction to Analysis SOLUTIONS
نویسنده
چکیده
1.9 Decide for which n the inequality 2 > n holds true, and prove it by mathematical induction. The inequality is false n = 2, 3, 4, and holds true for all other n ∈ N. Namely, it is true by inspection for n = 1, and the equality 2 = 4 holds true for n = 4. Thus, to prove the inequality for all n ≥ 5, it suffices to prove the following inductive step: For any n ≥ 4, if 2 ≥ n, then 2 > (n+ 1). This is not hard to see: 2 = 2 · 2 ≥ 2n, which is greater than (n+1) provided that (n+1) < √ 2n i.e. when n > 1/( √ 2−1) = √ 2+1, which includes all integers n ≥ 4.
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