A new elementary proof of the Prime Number Theorem

Let $\mathrm{\Omega }(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{C}$ one has $$\frac{1}{N}\sum _{n=1}^{N}f(\mathrm{\Omega }(n)+1)=\frac{1}{N}\sum _{n=1}^{N}f(\mathrm{\Omega }(n))+{\mathrm{o}}_{N\to \mathrm{\infty }}(1)\mathrm{.}$$ This yields a new elementary proof of the Prime Number Theorem.