Question

In the real line R, which of the following is true?
(A) Every sequence converges.
(B) Every Cauchy sequence converges.
(C) Every bounded sequence converges.
(D) None of the above.​

Answer 1

Answer:

Option number a is answer

Answer 2

A. In the real line, R is "every sequence converges".

Every sequence converges:

• A recurrence relation defines a convergent sequence.
• When the limit has been found to a precision of at least three decimal places, the calculation should come to an end.
• Give an example of a condition that could be utilized in this situation.
• Every convergent sequence is a sequence, that is the set   $\left \{ {xn : N} \right.$∈ N} is bounded.
• The condition given in the previous result is necessary but not sufficient. For example, the sequence ((−1)n) is a bounded sequence but it does not converge.

Every decreasing sequence convergent:

• The theorems say, informally, that if a sequence is increasing and is bounded above by a supremum, the series will converge to the supremum.
• Similarly, a decreasing sequence that is restricted below by an infimum will converge to the infimum.