UP TGT Mathematics Real Analysis Test Series - Part 5
Which of the following spaces is COMPLETE (i.e., every Cauchy sequence converges within the space)?
Type: MCQ • Marks: 1
Which of the following spaces is COMPLETE (i.e., every Cauchy sequence converges within the space)?
Options
A. Set of rational numbers Q
B. Set of irrational numbers
C. Set of real numbers R
D. Open interval (0, 1)
Explanation
Correct Answer: Option 3 (Set of real numbers R). 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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