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

Step 1: Understanding the Concept:
In mathematical analysis, a sequence \((x_n)\) is bounded if its terms lie within a finite interval.
A sequence is a Cauchy sequence if its terms become arbitrarily close to each other as the sequence progresses.
In the set of real numbers (\(\mathbb{R}\)), a sequence converges if and only if it is a Cauchy sequence (the completeness of \(\mathbb{R}\)).

Step 3: Detailed Explanation:
Let us evaluate each statement to identify the correct one:
- Evaluating Option (A): "Every bounded sequence is convergent."
This is false. For example, the sequence \(a_n = (-1)^n\) is bounded between \(-1\) and \(1\), but it oscillates and does not converge.
- Evaluating Option (B): "Every bounded sequence is a Cauchy sequence."
This is false. Because every Cauchy sequence in \(\mathbb{R}\) must converge, and some bounded sequences (like \((-1)^n\)) do not converge, not all bounded sequences are Cauchy.
- Evaluating Option (C) and (D):
By definition, a sequence \((x_n)\) is Cauchy if for every \(\epsilon > 0\), there exists an integer \(N\) such that:
\[ |x_n - x_m| < \epsilon \quad \text{for all } n, m \ge N \] If we choose \(\epsilon = 1\), there exists \(N\) such that:
\[ |x_n - x_N| < 1 \quad \text{for all } n \ge N \] This means all terms beyond \(N\) are bounded within the interval \((x_N - 1, x_N + 1)\).
The remaining finite set of terms before \(N\) (\(x_1, x_2, \dots, x_{N-1}\)) must also be finite.
Therefore, we can find a upper bound \(M\) for the entire sequence:
\[ M = \max(|x_1|, |x_2|, \dots, |x_{N-1}|, |x_N| + 1) \] This mathematical proof shows that every Cauchy sequence is bounded.
Thus, Option (C) is false and Option (D) is true.

Step 4: Final Answer:
The true statement is that every Cauchy sequence is bounded, matching Option (D).