Do you agree that ---> this is Betrand's conjecture (and it is not easy to prove, but it seems you are allowed to use it)?
Then by induction,
.
Thus, .
Let denote the nth prime. For n>3, show that
My proof so far:
Let
Now, 4 is in S because
Assume k in S, then
Now,
By the Bertrand's postulate, there exist at least one prime number q such that
Pick q to be the smallest prime from the set, then
But now I'm stuck, I'm trying to show that , how do I go about doing that?