# Examples of bounded sequence with infinite sub sequential limits.

Can anyone please help me to find an "Examples of bounded sequence with infinite sub sequential limits."

• A constant sequence is an easy example. – Mr Toad Mar 23 '17 at 5:29
• @MrToad Sir but a constant sequence has only one limit which is equal to all the sub-sequential limit. I am looking for a bounded sequence having infinitely different many sub-sequential limits. Suppose (-1)^n has only two sub-sequential limit namely "+1" and "-1" i.e only two sub-sequential limit but I want infinitely many. – Arkaprabha Karmakar Mar 23 '17 at 5:37
• A bijection from $\mathbb{N}$ to $\mathbb{Q}\cap[0,1]$? – delt3 Mar 23 '17 at 5:54
• Users delt3, angryavian and Mr Toad already provided nice examples. A much harder example is the sequence $\{ \sin n : n \in \Bbb{N} \}$. The set of limit points is all of the interval $[-1 ,1]$. – Sangchul Lee Mar 23 '17 at 6:12

You can enumerate all of the rationals in $[0,1]$. Then given any $1/n$ for $n \in \mathbb{Z} ^+$ (of which there are infinitely-many) there exists an infinite subsequence converging to that number.