- #1
B3NR4Y
Gold Member
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- 8
Homework Statement
Assume f:(a,b)→ℝ is differentiable on (a,b) and that |f'(x)| < 1 for all x in (a,b). Let an
be a sequence in (a,b) so that an→a. Show that the limit as n goes to infinity of f(an) exists.
Homework Equations
We've learned about the mean value theorem, and all of that fun stuff.
The Attempt at a Solution
I don't really know where to start so I brainstormed a couple of things I noticed[/B]
I know that since |f'(x)| is always less than one, any sequence of points will be bounded. Since they are bounded they are cauchy.
I also know that
## lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} ## exists for all x. I assume this should also mean that ## lim_{n \rightarrow \infty} \frac{f(a_n)-f(a)}{a_n - a} ## exists. I think with these two facts I can construct a proof but I don't know if either of the two are correct