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Explain why every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

Short Answer

Expert verified

If the sequence akis a monotonic sequence, then it is either increasing sequence or decreasing sequence.

If the sequence akis a decreasing sequence, then there exists a real number Msuch that M>ak, for every k. Hence, this sequence is bounded above.

If the sequence akis increasing sequence, then there exists a real number Msuch that M<akfor every k. 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.

Step by step solution

01

Step 1. Given information

We need to explain that every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

02

Step 2. Consider the sequence ak.

If the sequence akis 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 Msuch that M>akfor every k.

Hence, this sequence is bounded above.

Consider the sequence -n2n=0

The terms of the sequence are 0,-1,-4,....

The sequence is bounded above by 0.

Hence, the sequence -n2n=0is bounded above.
03

Step 3. If the sequence is increasing sequence, then there exists a real number M, such that M<ak for every k.

Hence, the sequence akis bounded below.

Consider the sequence 2nn=0

The terms of the sequence are, 1,4,8,....

The sequence is bounded above by 1.

Hence, the sequence2nn=0is bounded below.

04

Step 4. If the sequence is a bounded sequence, then it has both an upper bound and a lower bound.

Consider the sequence nn+1n=0

The above sequence is an increasing sequence, hence monotonic sequence.

The upper bound of the sequence is 1and the lower bound of the sequence is 0.

Thus, every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

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