Question

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 $\left\{a_k\right\}$is a monotonic sequence, then it is either increasing sequence or decreasing sequence.

If the sequence $\left\{a_k\right\}$is a decreasing sequence, then there exists a real number $M$such that $M>a_k$, for every $k$. Hence, this sequence is bounded above.

If the sequence $\left\{a_k\right\}$is increasing sequence, then there exists a real number $M$such that $M<a_k$for 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

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.

Step 2. Consider the sequence ak.

If the sequence $\left\{a_k\right\}$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 $M$such that $M>a_k$for every $k$.

Hence, this sequence is bounded above.

Consider the sequence ${\left\{-n^2\right\}}_{n=0}^{\infty }$

The terms of the sequence are $\left\{0,-1,-4,...\right\}$.

The sequence is bounded above by $0$.

Hence, the sequence ${\left\{-n^2\right\}}_{n=0}^{\infty }$is bounded above.

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 $\left\{a_k\right\}$is bounded below.

Consider the sequence ${\left\{2^n\right\}}_{n=0}^{\infty }$

The terms of the sequence are, $\left\{1,4,8,...\right\}$.

The sequence is bounded above by $1$.

Hence, the sequence${\left\{2^n\right\}}_{n=0}^{\infty }$is bounded below.

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

Consider the sequence ${\left\{\frac{n}{n+1}\right\}}_{n=0}^{\infty }$

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

The upper bound of the sequence is $1$and 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.