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4. Prove or disprove by giving a counterexample: (a) If (xn) and (Yn) are divergent sequences then (xn + Yn) diverges also. (b) If (xn) and (yn) are divergent sequences then (xn Yn) diverges also. (c) If (xn) and (¤n · Yn) are convergent sequences then (yn) converges also.

4. Prove or disprove by giving a counterexample: (a) If (xn) and (Yn) are divergent sequences then (xn + Yn) diverges also. (b) If (xn) and (yn) are divergent sequences then (xn Yn) diverges also. (c) If (xn) and (¤n · Yn) are convergent sequences then (yn) converges also.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Using basic real analysis 1, solve the question

**Exercise 4: Prove or Disprove by Giving a Counterexample** (a) If \((x_n)\) and \((y_n)\) are divergent sequences, then the sequence \((x_n + y_n)\) diverges also. (b) If \((x_n)\) and \((y_n)\) are divergent sequences, then the sequence \((x_n \cdot y_n)\) diverges also. (c) If \((x_n)\) and \((x_n, y_n)\) are convergent sequences, then the sequence \((y_n)\) converges also. *Instructions: For each statement, either prove the statement is true or provide a counterexample to demonstrate that it is false.*
Transcribed Image Text:**Exercise 4: Prove or Disprove by Giving a Counterexample** (a) If \((x_n)\) and \((y_n)\) are divergent sequences, then the sequence \((x_n + y_n)\) diverges also. (b) If \((x_n)\) and \((y_n)\) are divergent sequences, then the sequence \((x_n \cdot y_n)\) diverges also. (c) If \((x_n)\) and \((x_n, y_n)\) are convergent sequences, then the sequence \((y_n)\) converges also. *Instructions: For each statement, either prove the statement is true or provide a counterexample to demonstrate that it is false.*
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