Question

The necessary condition of a convergent sequence is :

• A. Bounded
• B. Having more than one limit
• C. Monotonic
• D. Not Cauchy

Hint:

Remember the logical hierarchy:
- Convergent \(\implies\) Bounded (Boundedness is necessary).
- Bounded \(\,\,\not\!\!\!\implies\) Convergent (e.g., \(a_n = (-1)^n\) is bounded but oscillates, so it is not convergent).
Thus, boundedness is necessary, but not sufficient.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A sequence of real numbers \(\{a_n\}\) is said to be convergent if it approaches a single, finite limit as \(n \to \infty\).
A necessary condition is a requirement that must be satisfied for the sequence to converge, though satisfying it does not guarantee convergence.

Step 2: Detailed Explanation:
Let us review the properties of convergent sequences:
- Boundedness: A sequence \(\{a_n\}\) is bounded if there exist real numbers \(M\) and \(m\) such that \(m \le a_n \le M\) for all \(n \in \mathbb{N}\).
According to a fundamental theorem of real analysis:
"Every convergent sequence of real numbers is bounded."
Therefore, boundedness is a necessary condition for convergence. If a sequence is unbounded, it cannot converge. This makes Option (A) correct.
Let us examine why the other options are incorrect:
- Option (B): "Having more than one limit." This is incorrect because a convergent sequence must have a unique limit.
- Option (C): "Monotonic." A sequence does not need to be monotonic to converge.
For example, the sequence \(a_n = \frac{(-1)^n}{n}\) oscillates and is not monotonic, yet it converges to 0. Thus, monotonicity is not a necessary condition.
- Option (D): "Not Cauchy." According to Cauchy's convergence criterion, a sequence of real numbers is convergent if and only if it is a Cauchy sequence. Thus, a convergent sequence must be Cauchy.
Therefore, only Option (A) is correct.

Step 3: Final Answer:
The correct option is (A).