Hi everyone,
I have struggled with this question for far too long and I am hopeful someone can help.
Let {x_{n}} be a sequence in R. Let 0<r<1 and suppose |x_{n+1}-x_{n|}<r^{n} for all n in N. I need to show {x_{n}} converges.
I know this can be done by showing {x_{n}} is a Cauchy sequence. I think I am on the right track for the first few steps. However, I quickly get lost. Please help.
Isn't the sequence in the problem strictly smaller than the sequence of partial sums of a geometric series, since 0 < r < 1? Therefore it is bounded.
Edit: actually it isn't necessarily either strictly increasing or strictly decreasing, since all we know is the absolute value of the difference of terms. But the sequence still has to converge, see if you can use the aforementioned fact to finish the proof.
I think the most natural way is to go with SworD's line of thought, esp. if this question is for an advanced calculus/analysis course.
A WLOG (without loss of generality) assumption that you could make is that the sequence is increasing hence monotonic, etc.
Hope this helps.
Update: You also need to mention the existence of a convergent subsequence due to the boundedness condition. Since you know that this sequence is Cauchy, what does a convergent subsequence mean for the whole sequence (knowing that the convergent subsequence is arbitrary)?
The hint we were given is this:
|x_{m}-x_{n}|=|x_{m}-x_{m+1}+x_{m+1}-x_{m+2}+...+x_{n-1}+x_{n}|
<=|x_{m}-x_{m+1}|+|x_{m+1}-x_{m+2}|+...+|x_{n-1}+x_{n}|
<=r^{m}+r^{m+1}+...+r^{n}
<=r^{m}(1+r+r^{2}+...r^{n-m})
It was given by another student and I'm not sure it is correct. If it is, I can finish up the proof. I'm just not convinced that this is the right direction to go with this.
What class is this for? I'm just curious about how much material (theorems, definitions, etc.) on sequences/convergence you're supposed to know.
If you can't invoke any theorems/lemmas, then use the hint and proceed.
Thanks for all the replies. I have a proof that I think is correct, but it is not the same as the proof in reply #10. This is a Real Number Analysis course. We are free to use any theorems or definitions that we have covered in class or in the homework. But, our selection is somewhat limited at this point. Thanks again for all the helpful replies.