Theorem 10: Prime Counting Function and Loglog x

[AlbertEinstein]

I am going through Hardy's book on number theory.The following theorem I do not understand.

theorem 10: pi[x] >= loglog x
where pi[x] is the prime counting function
and >= stands for greater than or equal to

The arguments written in the book are very compact.please help .

 

[shmoe]

Do you follow any of it?

Do you understand how they derived p_{n}<2^{2^n} ?

this is an important step. The rest just follows from pi(x) being increasing, and also \pi(p_n)=n which they use but don't explicitly mention.

 

[CRGreathouse]

Bertrand's postulate?

 

[shmoe]

Nope! the bound of p_n above is much weaker than Bertrand's will give you. It's correspondingly simpler to prove though, it follows from a slight adaptation of Euclid's proof there are infinitely many primes (in case anyone who hasn't seen it wants to give it a stab)