Question

In a metric space, is every Cauchy sequence necessarily a convergent sequence?

• A. Yes, always
• B. No, not always
• C. Only in finite spaces
• D. Only for bounded sequences

Hint:

Cauchy \( \Rightarrow \) Convergent only in complete spaces

Answer 1

The Correct Option is B

Concept: Cauchy sequence and completeness

• A sequence is Cauchy if its terms get arbitrarily close to each other.
• A sequence converges if it approaches a limit in the space.

Step 1: Key idea
In general metric spaces: \[ \text{Cauchy sequence} \not\Rightarrow \text{convergent} \]
Step 2: Special case
If the space is complete, then: \[ \text{Every Cauchy sequence converges} \]
Step 3: Example
In \( \mathbb{Q} \), a Cauchy sequence may converge to an irrational number, which is not in \( \mathbb{Q} \), so it does not converge in that space. Conclusion: \[ \text{Not every Cauchy sequence is convergent (unless space is complete)} \]