Second Chebyshev Function is Increasing

Theorem
The second Chebyshev function $\psi$ is increasing.

Proof
Let $x \ge y$.

Then:

From Logarithm is Strictly Increasing:


 * $\ln p \ge \ln 2 > 0$

So, we have:


 * $\ds \sum_{k \ge 1} \sum_{x < p^k \le y} \ln p \ge 0$

so:

So if $x \le y$, then:


 * $\map \psi x \le \map \psi y$

so:


 * $\psi$ is increasing.