Elementary Methods in Number Theory Exercise 1.5.11

Prove that $\pi(n)\leq \frac{n}{3}$ for $n\geq 33$.

Proof:According to Eratosthenes's sieve method, when $n\geq 33$,$\sqrt{33}\geq 5$.Then we delete all the multiples of the prime number 2(2 excluded),and all the multiples of prime number 3(3 excluded) and all the multiples of prime number 5(5 excluded),and 1.

\begin{equation}
\pi(n)\leq n-[\frac{n}{2}]-[\frac{n}{3}]-[\frac{n}{5}]+[\frac{n}{6}]+[\frac{n}{10}]+[\frac{n}{15}]+1+1+1-1\leq\frac{n}{3}
\end{equation}(Why?)