A sequence of real numbers converges if and only if it is not Cauchy. So one way to show that this sequence is not Cauchy is to show that it does not converge.
Try taking an arbitrary and then take and use them in the definition of a cauchy sequence (and think why this works)
You had me confused for a moment! The definition of Cauchy sequence is that with m and n going to infinity independently. You could not use m= n+1 to prove a sequence is Cauchy but it certainly can be used to give a counter example.
That's a very good point, and a good exercise for the thread creator would be to conjure up a counter example for the case - ie. a sequence for which the last limit is zero but the sequence does not converge.
That's a very good point, and a good exercise for the thread creator would be to conjure up a counter example for the case - ie. a sequence for which the last limit is zero but the sequence does not converge.