Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.

A bounded sequence

Short answer

A sequence $a_n$is a bounded sequence if it is bounded above and bounded below. If a sequence is not bounded, it is an unbounded sequence.

For example, the sequence ${\left(-1\right)}^n$is bounded.

Step-by-step solution

Step 1. Given Information  

The given statement is a bounded sequence.

Step 2. Explanation 

A sequence $a_n$is bounded aboveif there exists a real number M such that $a_n\le M$for all positive integers n.

A sequence $a_n$ is bounded below if there exists a real number M such that

$M\le a_n$for all positive integers n.

A sequence $a_n$is a bounded sequence if it is bounded above and bounded below. If a sequence is not bounded, it is an unbounded sequence.

For example, the sequence${\left(-1\right)}^n$is bounded.