Just as you said, .
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 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.
But the sum converges to 3/2. So .
Aren't we trying to show that |x_J - x_K| goes to 0?
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