Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Let $\langle x_n \rangle$ be a sequence in $\mathbb{R}$, where $x_n = \sin\left(\frac{1}{n}\right)$, then $\langle x_n \rangle$ is convergent in $\mathbb{R}$. Reason R : $\langle x_n \rangle$ is bounded.
In the light of the above statements, choose the correct answer from the options given below

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is NOT the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Hint:

Remember: Convergent $\implies$ Bounded, but Bounded $\centernot\implies$ Convergent! Counterexample: $x_n = (-1)^n$ is bounded in $[-1, 1]$ but oscillates and diverges.

Answer 1

The Correct Option is B

Step 1: Concept
A sequence $\langle x_n \rangle$ is convergent if $\lim_{n \to \infty} x_n = L$ for some finite $L \in \mathbb{R}$. Every convergent sequence is bounded, but boundedness alone is not sufficient to guarantee convergence (e.g., $y_n = (-1)^n$ is bounded but divergent).

Step 2: Key Formulas and Approach
1. Evaluate $\lim_{n \to \infty} \sin(1/n)$ using continuity of $\sin x$. 2. Check if $|x_n| \leq M$ for all $n \in \mathbb{N}$. 3. Determine whether Reason R is a sufficient logical explanation for Assertion A.

Step 3: Step-by-step Explanation

• Evaluating Assertion A: As $n \to \infty$, $1/n \to 0$. By continuity of the sine function: \[ \lim_{n \to \infty} \sin\left(\frac{1}{n}\right) = \sin\left(\lim_{n \to \infty} \frac{1}{n}\right) = \sin(0) = 0 \] Since the limit exists and is finite, the sequence converges to $0$. Hence, Assertion A is true.
• Evaluating Reason R: For all $n \geq 1$, $|\sin(1/n)| \leq 1$. Thus, the sequence $\langle x_n \rangle$ is bounded. Hence, Reason R is true.
• Evaluating Logical Explanation: Boundedness does not guarantee convergence of a sequence. To guarantee convergence, a sequence must be both bounded and monotonic (Monotone Convergence Theorem). Therefore, Reason R is a true statement, but NOT the correct explanation of Assertion A.

Step 4: Final Answer
Both A and R are true, but R is NOT the correct explanation of A. Thus, Option (B) is correct.