I am writing an extended piece of work and, without wanting to assume the prime number theorem, I have to make the replacement

$\displaystyle O\big(\psi(x)\big) = O(x)$;

that is, I want to show $\displaystyle \psi(x) = O(x)$, where as usual $\displaystyle \psi(x) =\sum_{n\leq x} \Lambda(n)$. Every proof I have seen of this sort gets a little too involved (almost proving the PNT) and I really cannot see how this is not easily shown. I mean, it is nowhere near as strong as PNT. Can anyone show a simpler way?