TITLE: Prove that the Dirichlet eta function is monotonic
QUESTION [10 upvotes]: Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By elementary I mean that this is a problem of real analysis, and there must be a solution without using analytic continuation for $\zeta(p)$ and its relations to $\eta(p)$. Another notable remark is that function $g(p):=\frac{1}{a^p}-\frac{1}{(a+1)^p}$ ($a\geq1$) can be both increasing or decreasing at $p=p_0$, depending on $a$, which makes this problem more complex than it seems at first glance. Thanks in advance!

REPLY [4 votes]: This is Theorem 3 (p. 10) in the Report:
J. van de Lune,  Some inequalities involving Riemann's zeta-function,
CWI Report  ZW 50/75, CentruM Wiskunde & Informatica, Amsterdam,1975,
in which the proof does not use complex variables. The pdf of this 
Report can be found in the CWI repository
https://repository.cwi.nl/noauth/search/fullrecord.php?publnr=6895