Chapter 8, Problem 9RE

1. (Requires a little knowledge of calculus) Show that $1+\frac{1}{2}+\frac{1}{3}+\mathrm{.....}+\frac{1}{n}=O\left(\log n\right)$

2. Show that the Sieve of Eratosthenes, used to find all primes less than or equal to a given integer n, can be implemented with an algorithm that is $O\left(n\log n\right)$. Count comparisons and crossing out an integer as basic operations.

(See Section 4.3 and especially paragraph 4.3.5)