Question

Which of the following statement is true for sequence of real number

• A. Every bounded sequence is convergent
• B. Every bounded sequence is Cauchy sequence
• C. Every Cauchy sequence may not bounded
• D. Every Cauchy sequence is bounded

Hint:

Remember the core relationships in \(\mathbb{R}\):
- Cauchy \(\iff\) Convergent (always equivalent in complete spaces).
- Convergent \(\implies\) Bounded (always true).
- Therefore, Cauchy \(\implies\) Bounded (always true).
- The reverse is not always true: Bounded sequences do not always converge.

Answer 1

The Correct Option is D