<s>[INST] Find the cluster points and one convergent subsequence of $a_n = \sqrt{n+1}-\sqrt{n} -\frac{(-1)^n}{n}, n \in \Bbb N$ Find the cluster points and one convergent subsequence of $a_n = \sqrt{n+1}-\sqrt{n} -\frac{(-1)^n}{n}, n \in \Bbb N$
For large $n$ would this not converge to $0$ and the only cluster point be also $0$? 
Thus, any subsequence of $\{a_n\}$ would also converge to $0$?
Is this correct or have I missed something here?
 [/INST] $$\sqrt{n+1} - \sqrt{n} - \frac{(-1)^n}{n} = \sqrt{n} \left( \sqrt{1+ \frac{1}{n}} - 1\right) - \frac{(-1)^n}{n}$$
$$= \sqrt{n} \left( \frac{1}{2n} + o \left( \frac{1}{n}\right)\right) - \frac{(-1)^n}{n}= \frac{1}{2 \sqrt{n}} - \frac{(-1)^n}{n} + o \left( \frac{1}{\sqrt{n}}\right)$$
So the sequence tends to $0$, so there is only one cluster point ($0$), and all the subsequences tend to $0$ as well.
</s>