Question

Which of the following statements is/are TRUE?

• A. There exists a monotone sequence that does not converge but has a convergent subsequence.
• B. There exists a sequence that has a bounded subsequence but does not have any convergent subsequence.
• C. There exists a sequence \( (x_n) \) such that given any positive integer \( m \), \( (x_n) \) has a subsequence converging to \( m \).
• D. There exists a sequence \( (x_n) \) such that \( (|x_{n+1} - x_n|) \) converges to 0 but \( (x_n) \) does not converge.

Hint:

The Bolzano-Weierstrass theorem guarantees that every bounded sequence has a convergent subsequence. This is useful in analyzing the behavior of sequences and their subsequences.

Answer 1

The Correct Option is C, D

Step 1: Option A, monotone but no limit.
If a monotone sequence does not converge, it must run off to $+\infty$ or $-\infty$. Then every subsequence also runs off the same way, so it can have no convergent subsequence. So A is false.

Step 2: Option B, bounded part with no limit point.
By the Bolzano Weierstrass theorem any bounded sequence has a convergent subsequence. So a bounded subsequence must itself contain a convergent piece. Option B is false.

Step 3: Option C, hitting every integer.
List the terms as $1,1,2,1,2,3,1,2,3,4,\dots$ so each positive integer appears infinitely often. Then for any $m$ we can pick the terms equal to $m$ and get a subsequence going to $m$. Option C is true.

Step 4: Option D, small steps but no limit.
Take $x_n=\sqrt{n}$. Then $x_{n+1}-x_n=\sqrt{n+1}-\sqrt{n}=\frac{1}{\sqrt{n+1}+\sqrt{n}}\to 0$, yet $x_n\to\infty$. So the steps shrink while the sequence still does not settle. Option D is true.

Step 5: Conclusion.
The true statements are C and D.
\[ \boxed{C,\ D} \]