Assertion A: Convergence Analysis

We analyze the sequence $\langle x_n \rangle$ defined by $x_n = \sin\left(\frac{1}{n}\right)$ for convergence in $\mathbb{R}$.

• As $n$ approaches infinity ($n \to \infty$), the argument $\frac{1}{n}$ approaches 0.
• Because the sine function, $\sin(x)$, is continuous at $x=0$, we have: $ \lim_{n \to \infty} x_n = \lim_{n \to \infty} \sin\left(\frac{1}{n}\right) = \sin\left(\lim_{n \to \infty} \frac{1}{n}\right) = \sin(0) = 0 $
• Since the limit exists and is a finite real number ($0$), the sequence $\langle x_n \rangle$ is convergent.

Therefore, Assertion A is true.

Reason R: Boundedness Analysis

We determine if the sequence $\langle x_n \rangle = \left\langle \sin\left(\frac{1}{n}\right) \right\rangle$ is bounded.

• For all integers $n \ge 1$, the term $\frac{1}{n}$ satisfies $0 < \frac{1}{n} \le 1$.
• The sine function outputs values between -1 and 1. Specifically, for $x$ in the interval $(0, 1]$, $\sin(x)$ is positive and less than or equal to $\sin(1)$ (which is approximately 0.841).
• Thus, for all $n \ge 1$, we have $0 < x_n = \sin\left(\frac{1}{n}\right) \le \sin(1)$.
• Since the sequence terms are confined within the interval $(0, \sin(1)]$, the sequence is bounded below by 0 and bounded above by $\sin(1)$.

Therefore, Reason R is true.

Relationship Between Assertion A and Reason R

We assess if Reason R is the correct explanation for Assertion A.

• Boundedness is a necessary condition for convergence in $\mathbb{R}$, but it is not sufficient. A sequence can be bounded without converging (e.g., $\langle (-1)^n \rangle$).
• The convergence of $\langle \sin(1/n) \rangle$ is primarily explained by the limit of its argument ($\frac{1}{n} \to 0$) combined with the continuity of the sine function at 0.
• While the sequence is indeed bounded, this property alone does not guarantee convergence. The reason for convergence is the limit approaching $\sin(0)$.

Hence, Reason R is true, but it is not the correct explanation for Assertion A.

Final Conclusion

Based on the analysis, Assertion A is true, and Reason R is true. However, Reason R does not provide the correct explanation for why Assertion A is true.

The correct choice is: Both A and R are true but R is NOT the correct explanation of A.