UP TGT Mathematics Real Analysis Test Series - Part 2
Is the condition |a_(n+1) - a_n| -> 0 as n -> infinity SUFFICIENT for a sequence to be Cauchy?
Type: MCQ • Marks: 1
Is the condition |a_(n+1) - a_n| -> 0 as n -> infinity SUFFICIENT for a sequence to be Cauchy?
Options
A. Yes, always sufficient
B. Yes, if positive
C. No, counterexample a_n = 1 + 1/2 + ... + 1/n
D. Yes, for all real sequences
Explanation
Correct Answer: Option 3 (No, counterexample a_n = 1 + 1/2 + ... + 1/n). Cauchy General Principle: In R, a sequence converges if and only if it is a Cauchy sequence (|a_n - a_m| < epsilon for n, m >= N). Every Cauchy sequence in R is bounded and convergent.
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