Exercise.
Suppose that (
)
is bounded,
monotonic nondecreasing. Show that it is Cauchy. Hint: let
0
and take
such that
+
for all
.
(If there is no such , let
0
,
1
0
+
,
2
2
+
, and so on. Hence by AP show that
(
)
is unbounded.) Now prove directly that for
arbitrary ,
(
,
)
.