Let
be differentiable on
with
. Select
and define
for
. Prove that
is a convergent sequence.
Hint: To show is Cauchy, first show that for .
Okay, so it's pretty clear that I have to use the mean value theorem and so there exists
such that
for
. Now it seems that I should just replace
with
and I will arrive at the hint. However, how can I guarantee that for any
,
? Is there something I am missing that guarantees this, or is my actual approach incorrect so far? And even if I arrive at the hint, how would I proceed exactly? Thanks.