Can someone tell me if anything is wrong with this proof? It seems too good to be true, as it was very easy to create.
$$ \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} =c $$ $$ \ln(x!)=\psi(x)+\psi(x/2)+\psi(x/3)+\psi(x/4)...\psi(x/x) $$ $$ \frac{\ln(x!)}{\psi(x)+\psi(x/2)+\psi(x/3)+\psi(x/4)...\psi(x/x)}=1$$ $$ \frac{\ln(x!)/x}{(\psi(x)+\psi(x/2)+\psi(x/3)+\psi(x/4)...\psi(x/x) )/x}=1$$ $$ \lim_{x \rightarrow \infty}\frac{\ln(x!)/x}{(\psi(x)+\psi(x/2)+\psi(x/3)+\psi(x/4)...\psi(x/x))/x}=1$$ $$ \lim_{x \rightarrow \infty}\frac{\ln(x!)/x}{(c+c/2+c/3+c/4+c/5...c/x)}=1$$ $$ \lim_{x \rightarrow \infty}\frac{\ln(x!)/x}{c \times \text{harmonic}(x)}=1$$ $$ \lim_{x \rightarrow \infty}\frac{\ln(x!)/x}{\text{harmonic}(x)}=1/c$$ $$ c=1$$
$\psi(x)$ is the first chebyshev function identity and can be found here.