1. ## real analysis

$\displaystyle {prove \ that \ if \ \mid x_{n+1} - x_n \mid \leq \frac{1}{3^n} , \forall n \in N ,\ then \ x_n \ is \ a \ cauchy \ sequence\ .}$

2. If $\displaystyle J>K$ then $\displaystyle \left| {x_J - x_K } \right| \leqslant \sum\limits_{k = K}^{J - 1} {\left| {x_{k + 1} - x_k } \right|} \leqslant \sum\limits_{k = K}^{J - 1} {\frac{1}{{3^k }}}$

3. Just as you said, $\displaystyle |x_J - x_K| \le \sum_{k = K}^{J-1} \frac{1}{3^k}$.

That's a finite sum, and so for large enough k, the entire sum should go to 0. But if we're trying to prove that $\displaystyle |x_J - x_K| < \epsilon$ for all J, K > k, then isn't it possible to hold K fixed and take J so large that the sum doesn't actually approach 0?

Let me know if my question isn't clear enough.

4. Originally Posted by JG89
Just as you said, $\displaystyle |x_J - x_K| \le \sum_{k = K}^{J-1} \frac{1}{3^k}$.
That's a finite sum, and so for large enough k, the entire sum should go to 0. But if we're trying to prove that $\displaystyle |x_J - x_K| < \epsilon$ for all J, K > k, then isn't it possible to hold K fixed and take J so large that the sum doesn't actually approach 0?
But the series $\displaystyle \sum_{k = 1}^{\infty} \frac{1}{3^k}$ converges.
Therefore the sequence of partial sums converges; so it is a Cauchy sequence.
That means we done.

5. But the sum converges to 3/2. So $\displaystyle |x_J - x_K| \le 3/2$.

Aren't we trying to show that |x_J - x_K| goes to 0?

6. Originally Posted by JG89
But the sum converges to 3/2. So $\displaystyle |x_J - x_K| \le 3/2$.
Aren't we trying to show that |x_J - x_K| goes to 0?
No, you have it all wrong.
$\displaystyle \sum\limits_{k = K}^{J} {\frac{1}{3^k}}$ is the difference in the partial sums, $\displaystyle S_J~\&~S_{K+1}$.
Because $\displaystyle \left( {S_n } \right)$ is a Cauchy sequence that difference can be made as small as necessary.

7. Wow, I had worked that out but didn't realize I was working out the partial sums...no excuses though, that was a silly mistake. Thanks for the help Plato