I have been given the following question in my homework:
Show that for every $n \in \mathbb{N}$ there are at least ${\log_2log_2 (n-1)}$ primes in $[1,n]$
I think it is related to euclidean and induction but I am having difficulties developing my solution.
Thanks for your help (:
Euclid's proof that there are infinitely many primes also shows that the $n$-th prime satisfies $p_n\le 2^{2^{n}}$, which also gives $\pi(n)\ge \log_2(\log_2(n-1))$ , see this MSE question.
Remark: We can easily prove a better estimate, using an elementary argument starting with $2^n\le \binom{2n}{n}<4^n$, which gives $$ \pi(n)>\frac{n}{6\log n} $$ for all $n\ge 2$ (see also this MSE question).
