UP TGT Mathematics Real Analysis Test Series - Part 2
A sequence (a_n) is called a contractive sequence if there exists a constant 0 < C < 1 such that for all n:
Type: MCQ • Marks: 1
A sequence (a_n) is called a contractive sequence if there exists a constant 0 < C < 1 such that for all n:
Options
A. |a_n| <= C
B. |a_(n+1) - a_n| < C
C. |a_(n+1)| <= C * |a_n|
D. |a_(n+2) - a_(n+1)| <= C * |a_(n+1) - a_n|
Explanation
Correct Answer: Option 4 (|a_(n+2) - a_(n+1)| <= C * |a_(n+1) - a_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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