Chapter 7: Q. 18 (page 592)
Explain why every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.
Short Answer
If the sequence is a monotonic sequence, then it is either increasing sequence or decreasing sequence.
If the sequence is a decreasing sequence, then there exists a real number such that , for every . Hence, this sequence is bounded above.
If the sequence is increasing sequence, then there exists a real number such that for every . Hence, this sequence is bounded below.
If the sequence is a bounded sequence, then it has both an upper bound and a lower bound.
Therefore, every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.