How can we show that the sequence $\displaystyle (-1)^{n+1}$is bounded but isnot a Cauchy Sequence?
You had me confused for a moment! The definition of Cauchy sequence is that $\displaystyle \lim_{m,n\to 0} |a_n- a_m|= 0$ 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 $\displaystyle \lim_{n\to \infty} |a_{n+1}- a_n|= 0$ - ie. a sequence for which the last limit is zero but the sequence does not converge.