For each positive real number lambda. Let A_l | vedclass

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Question

(c)

We have,

$|\sin \sqrt{n+1}-\sin \sqrt{n}| < \lambda, \lambda \in R$

When $n \rightarrow$

$|\sin \sqrt{n+1}-\sin \sqrt{n}| \rightarr

Vedclass · Accepted Answer

$A_{12}^c, A_{13}^c, A_{25}^c$ are all finites sets

Vedclass · Answer

(c)

We have,

$|\sin \sqrt{n+1}-\sin \sqrt{n}| < \lambda, \lambda \in R$

When $n ightarrow$

$|\sin \sqrt{n+1}-\sin \sqrt{n}| ightarrow 0$

$	herefore$ There exist infinite natural number for which $|\sin \sqrt{n+1}-\sin \sqrt{n}| < \lambda, \forall \gamma \lambda \in 0$

Hence, $A_{1 / 2}, A_{1 / 3}, A_{2 / 5}$ are infinite sets. $	herefore A_{1 / 2}^e, A_{1 / 3}^e, A_{2 / 5}^e$ are finite sets.

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

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