Question

For each positive real number $\lambda$, let $A_{\lambda }$ be the set of all natural numbers $n$ such that $|\sin \left(\sqrt{n-1}\right)–\sin \left(\sqrt{n}\right)|<\lambda$. Let $A_{\lambda }^c$ be the complement of $A_{\lambda }$ in the set of all natural numbers. Then

• A. $A_{\frac{1}{2}},A_{\frac{1}{3}},A_{\frac{2}{5}}$ are all finite sets
• B. $A_{\frac{1}{2}}\text{is a finite set but}{A}_{\frac{1}{3}},A_{\frac{2}{5}}$ are infinite sets
• C. $A_{\frac{1}{2}}^c,A_{\frac{1}{3}}^c,A_{\frac{2}{5}}^c$ are all finite sets
• D. $A_{\frac{1}{3}},A_{\frac{2}{5}}\text{are finite set but}{A}_{\frac{1}{2}}$ is an infinite set

Answer 1

The correct option is C $A_{\frac{1}{2}}^c,A_{\frac{1}{3}}^c,A_{\frac{2}{5}}^c$ are all finite sets

As $n\to \infty$,
$|\sin \left(\sqrt{n-1}\right)–\sin \left(\sqrt{n}\right)|\to 0$
Therefore, there can infinite number of values of $\lambda$ such that
$|\sin \left(\sqrt{n-1}\right)–\sin \left(\sqrt{n}\right)|<\lambda$, where $\lambda >0$
$\therefore A_{\frac{1}{2}},A_{\frac{1}{3}},A_{\frac{2}{5}}$ are all infinite sets
$\Rightarrow A_{\frac{1}{2}}^c,A_{\frac{1}{3}}^c,A_{\frac{2}{5}}^c$ are all finite sets