Question

Establish which of the following statements are true. (a) A sequence is convergent if and only if all of its subsequences are convergent. (b) A sequence is bounded if and only if all of its subsequences are bounded. (c) A sequence is monotonic if and only if all of its subsequences are monotonic. (d) A sequence is divergent if and only if all of its subsequences are divergent.

Answer 1

Statement A - True.

Statement B - False.

Statement C - True.

Statement D - False.

Explanation:

(a) A sequence is convergent if and only if all of its subsequences are convergent - this statement is correct.

(b) A sequence is bounded if and only if all of its subsequences are bounded - this statement is incorrect.

(c) A sequence is monotonic if and only if all of its subsequences are monotonic - this statement is correct.

(d) A sequence is divergent if and only if all of its subsequences are divergent - this statement is incorrect.