This is a question from The Art and Craft of Problem Solving.
Use the ideas from Problem 9.2.37 and Example 9.4.7 to show that, for any positive integer n, prove that the sum of the reciprocals of the prime numbers that do not exceed n is greater than ln(ln(n)) - 1/2.
This is Problem 9.2.37
$$ 9.2.37. \sum_{\text{n}=0}^{\infty}{\text{a}_{\text{n}}}\,\,\text{converges iff }\prod_{\text{n}=0}^{\infty}{\left( 1+\text{a}_{\text{n}} \right)}\,\,\text{converges}. $$
I have proved this question, by trimming them by e^S.
And this is 9.4.7, an easy one.
9.4.7 Prove that the number of primes is infinite, by considering zeta's function and harmonic series.
Obvious.
So my question is, how to combine ideas in these two questions, to prove that "The sum of the reciprocals of the prime numbers that do not exceed n is greater than ln(ln(n)) - 1/2"?