Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?
Solution 1:
Even if the sequence is bounded, the condition does not imply that the sequence is Cauchy.
Consider the following sequence: $$ 0,1,\frac12,0,\frac14,\frac12,\frac34,1,\frac78,\frac68,\frac58,\frac48,\frac38,\frac28,\frac18,0,\frac1{16},\ldots $$ The sequence goes back and forth between $0$ and $1$ in smaller and smaller steps. So $\lim(a_{n+1}-a_n)=0$, while the sequence oscillates between $0$ and $1$ and so it is not Cauchy.