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.
Okay, but I'm afraid of that there exists sequences that repeat indefinitely and therefore do not converge. However after constructing a few examples it seems that if it does repeat then the derivative at at least one point is greater than or equal to one. Is there a way to definitely prove this/is it not really important/does it come up in the proof/whatever? Thanks.
Also, how do I proceed with the hint exactly?