Find the cluster points and one convergent subsequence of $a_n = \sqrt{n+1}-\sqrt{n} -\frac{(-1)^n}{n}, n \in \Bbb N$

Question

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?

Answer 1

We have $$\sqrt{n+1} - \sqrt{n} = \frac1{\sqrt{n+1} + \sqrt{n}} \xrightarrow{n\to\infty} 0$$ and $$\frac{(-1)^n}n \xrightarrow{n\to\infty} 0$$ so $a_n \to 0$ as a difference of two convergent sequences. Hence $0$ is the only cluster point.

Answer 2

$$\sqrt{n+1} - \sqrt{n} - \frac{(-1)^n}{n} = \sqrt{n} \left( \sqrt{1+ \frac{1}{n}} - 1\right) - \frac{(-1)^n}{n}$$ $$= \sqrt{n} \left( \frac{1}{2n} + o \left( \frac{1}{n}\right)\right) - \frac{(-1)^n}{n}= \frac{1}{2 \sqrt{n}} - \frac{(-1)^n}{n} + o \left( \frac{1}{\sqrt{n}}\right)$$

So the sequence tends to $0$, so there is only one cluster point ($0$), and all the subsequences tend to $0$ as well.