Easy proof of falsehood of $\pi(n) \leq C \cdot \text{ln}(n)$ for the prime counting function $\pi$

Question

Let $\pi(n)$ be the number of primes in the range $1,\dotsc,n$.

The following statement is true: There is no $C>0$ such that $\pi(n) \leq C \cdot \text{ln}(n)$ for all $n\geq 1$.

It follows immediately from the prime number theorem which is a much stronger result.

Still, since the above statement is much weaker than the PNT, I was wondering if it has a simple proof.

Is there a proof of the above theorem which is simpler than the known proofs of the prime number theorem?

Answer 1

There are all kinds of rougher estimates, for example you can use Chebyshev's estimate to show that $\pi (x) > c x/ \log (x) $ for a positive c and large enough x.