Hi, I think I am close to the answer for this but I am missing something:
Suppose is a sequence satisfying for all n. Show that is a convergent sequence.
So to prove it is convergent, I need to prove it is a Cauchy sequence, ie for all there is an N such that for all .
Now and the triangle inequality gives .
Then using the inequality given in the question I get .
This is where I get stuck, I think I need to get rid of that m-n somehow.
Yes, you are quite close to the answer, right up to the inequality (where m>n). Here, instead of estimating each individual term (replacing each term by the largest term ), write the terms in the reverse order . This is a (finite) geometric series, whose sum is less than the infinite sum .
Ah cheers, I never think of using geometric series!