Can anyone provide an example of a sequence that satisfies the following definition, but it not Cauchy:
for every e>0 there exists natural number N such that for every n>or=N |xn-xn+1|<e
How about the sequence of partial sums $\displaystyle H_n=\sum_{k=1}^{n}\frac{1}{k}$. It's not Cauchy since it's divergent, but $\displaystyle H_{n+1}-H_n=\frac{1}{n+1}$, which can be made as small as desired.