Just to clarify, you're trying to show that
converges, correct? And you're required to use the Cauchy criterion?
What have you done so far?
How about this then...
Choose and write:
We need to show that for all we can find such that for all and we have
We can write:
since for all we may assume (with triangle inequality)
Now it can't be too hard to show that we can find such that for all and we have