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.