# An example of a bounded pseudo Cauchy sequence that diverges? [duplicate]

Harmonic series diverges and pseudo Cauchy however it's not bounded. So how can I find such a sequence?

A sequence $(s_n)$ is pseudo-Cauchy if, for all $\xi>0$, there exists an $N$ such that if $n ≥ N$, then $|s_{n+1}−s_n| < ξ$.

• What is a pseudo-Cauchy sequence? – user75930 Oct 16 '13 at 17:39
• $$\textstyle 0,\,{1\over 2},\, 1,\, {2\over 3},\,{1\over 3},\,{0},\,{1\over 4},\,{2\over 4},\,{3\over 4},\,1,\,{4\over 5},\,{3\over 5},\,{2\over 5},\,{1\over 5}, \,0, \ldots$$ – David Mitra Oct 16 '13 at 17:42
• A sequence (sn) is pseudo-Cauchy if, for all ξ> 0, there exists an N such that if n ≥ N, then |sn+1−sn| < ξ – user11111 Oct 16 '13 at 17:42

Try the sequence $$a_n=\sin\sqrt n$$
• @user11111 $\sqrt{n+1}-\sqrt n\rightarrow 0$ as $n\rightarrow\infty$. Nicer than my example. – David Mitra Oct 16 '13 at 18:01
• @user11111 By the Mean Value Theorem, $|\sin\sqrt{n+1}-\sin\sqrt n|\le|\sqrt{n+1}- \sqrt n|{\rightarrow} 0$. – David Mitra Oct 16 '13 at 18:18