UP TGT Mathematics Real Analysis Test Series - Part 5
A real sequence (a_n) is called a Cauchy sequence if for every epsilon > 0, there exists a positive integer N such that...
Type: MCQ • Marks: 1
A real sequence (a_n) is called a Cauchy sequence if for every epsilon > 0, there exists a positive integer N such that for all m, n >= N:
Options
A. |a_n - a_m| < epsilon
B. |a_n| < epsilon
C. |a_n - a_1| < epsilon
D. |a_n - L| < epsilon
Explanation
Correct Answer: Option 1 (|a_n - a_m| < epsilon). 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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