Let π(N) be the number of primes less than or equal to N (example: π(100) = 25). The famous prime number theorem then states (with ∼ meaning asymptotically equal): π(N) ∼N/log(N) Proving this theorem...

Let π(N) be the number of primes less than or equal to N (example: π(100) = 25). The famous prime number theorem then states (with ∼ meaning asymptotically equal):
                               π(N) ∼N/log(N)
Proving this theorem is very hard. However, we can derive a statistical form of the prime number theorem. For this, we consider random primes which are generated as follows:
(i) Create a list of consecutive integers from 2 to N.
(ii) Start with 2 and mark every number > 2 with a probability of 1/2.
(iii) Let n be the next non-marked number. Mark every number > n with a probability of 1/n.
(iv) Repeat (iii) until you have reached N.
All the non-marked numbers in the list are called random primes.
 Questions:

(a) Let q_n be the probability of n being selected as a random prime during this algorithm.
Find an expression for q_n
in terms of q_n-1.
(b) Prove the following inequality of q_n
and q_n+1:
                   1/q_n
+ 1/n <>_n+1 <>_n
+ 1/n − 1