From Littlewood's 1914 theorem (paraphrase):
I propose to show there are arbitrarily large values of x for which successively
$\psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A}$
$ \psi(x)- x > K \sqrt{x}\log\log\log x \tag{B}$
[...2 pages later]
It suffices then, to establish (A), (B), to show that a supposition such that
$\psi(x) -1 < \delta\sqrt{x}\log\log\log x \text{ for all } \delta >0 \text{ and } x > x_0(\delta),\tag{2}$
contradicts our [auxiliary lemma which seems to establish A, B].
My threshold question is whether (2) shouldn't read:
$|\psi(x) - x | < \delta \sqrt{x} \log\log\log x$?
Even if we amend (2) to
$\psi(x) - x < \delta \sqrt{x}\log\log\log x$
for a contradiction we could still have B but not A.
On the other hand assuming typos is a last resort when parsing a proof. It is good to see the overall strategy of the proof before working through it and (2) makes no sense to me. There is a brief account of the proof in the Wiki page (bottom) on Skewes' number but I think it casts no light on this question.
If someone happens to be familiar with the argument of the proof and can see how (2) as written (or otherwise) fits into it I would appreciate any help. Thank you. Unfortunately I do not see a link to the proof online and am using a borrowed hard copy.
It does look like that "$\psi(x) -1$" should be "$\psi(x) -x$". Since $\psi(x) \approx x$, $\psi(x) - 1$ is still $\approx x$.
Do you have a link to Littlewood's paper?