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.