<s>[INST] n such that $|\sin (\sqrt{n+1})-\sin \sqrt n|< \lambda$ 
I broke $|\sin (\sqrt{n+1})-\sin \sqrt n|$ as $|2cos(\frac{\sqrt{n+1}+\sqrt n}{2})sin(\frac{\sqrt{n+1}-\sqrt n}{2})|$.I am facing trouble in proving that it is less than some number for some n.Please help me in his regard.Thanks.
 [/INST] Note that for $0< x < \pi/2$, $\sin x < x$.
Also, for all $n > 0$,  we have:
$$
\frac{1}{4n} + 1 + n > n+1 \\
\left( \frac{1}{2\sqrt{n}}+\sqrt{n}\right)^2 > n+1 > 0 \\
\frac{1}{2\sqrt{n}} +\sqrt{n}> \sqrt{n+1} \\
\sqrt{n+1}-\sqrt{n}<\frac{1}{2\sqrt{n}}
$$
Also it is obvious that 
$$
\left|\cos \left( \frac{\sqrt{n+1}+\sqrt{n}}{2}\right)\right| \leq 1
$$
So for any $\lambda > 0$ and sufficiently large $n$,
$$
|\sin \sqrt{n+1}-\sin\sqrt{n} |< \lambda$$
Thus the answer is C.
</s>