Assume that each number k is crossed off by the Sieve of Eratosthenes every time a divisor of it is found. (For example, 6 is crossed off when 2 is the prime in question, and when 3 is the prime in...

Assume that each number k is crossed off by the Sieve of Eratosthenes every time a divisor of it is found. (For example, 6 is crossed off when 2 is the prime in question, and when 3 is the prime in question.) Prove that the total number of crossings-out by sieve(n) is ≤ Hn n, where Hn is the nth harmonic number. (See Definition 5.4.)