1.17. The basic argument in the proof of Theorem 8.3.2 was the "last time in (-E, E)". A harder but instructive argument usmg the "first time" may be given as follows

Show that for 1 :s m M M M L gn(E) :s L f~m) Lgn(2E).
Il=m n=m ¥
This fonn of condition (iii') is due to Hsu Pei; see also Chung and Lindvall, Proc. Amer. Math.
Soc. Vol. 78 (1980), p. 285.

It follows by a form of Lemma 1 in this section that now use Theorem 8.1.4.
x 1 lim '" f(m) > _. m-+x~ n -4'
n=m

fm P{|S;e for m j n -1; |S|