1.Generalize Theorem 5.4 to numbers that aren’t necessarily exact powers of 2. Specifically, prove that log n + 2 ≥ Hn ≥ (log n − 1)/2 + 1 for any real number n ≥ 1. (Hint: use Theorem 5.4.) 2. Prove...

1.Generalize Theorem 5.4 to numbers that aren’t necessarily exact powers of 2. Specifically, prove that log n + 2 ≥ Hn ≥ (log n − 1)/2 + 1 for any real number n ≥ 1. (Hint: use Theorem 5.4.)

2. Prove Bernoulli’s inequality: let x ≥ −1 be an arbitrary real number. Prove by induction on n that (1 + x) n ≥ 1 + nx for any positive integer n.