Chapter 8.1, Problem 53E

To determine

To explain: Why the sequence has a limit and say about the value of the limit.

Explanation of Solution

Given information: $\left\{a_n\right\}$is a decreasing sequence and all its terms lie between the numbers $5$ and $8$ .

Theorem1: if a sequence $\left\{a_n\right\}$ of real number is monotone decreasing and bounded below then it converges to its exact lower bound.

Definition1 : A ${\left\{a_n\right\}}_n$ is said to be bounded below if $\exists$ a real number $L\text{ s}\text{.t, }{x}_n\ge L,\forall n\in \mathbb{N}.L$ is then called a lower bound of ${\left\{a_n\right\}}_n$

Definition2:A sequence $\left\{a_n\right\}$ of areal numbers is said to be convergent if it has a finite number l as its limit. We then say that the sequence ${\left\{a_n\right\}}_n$ converges to l . Thus ${\left\{a_n\right\}}_n$ converges to l $\Rightarrow$ given $\epsilon >0;\exists n_0\in \mathbb{N}\text{ s}\text{.t}\text{. }\left|x_n-l\right|<\epsilon ,\forall n\ge n_0.$

Definition3: A sequence ${\left\{a_n\right\}}_n$ of real numbers is said to tend to a finite real number l, iff for any given positive number $\epsilon$ there exists a positive $n_0$ such that $\forall n\ge n_0,$ the inequality $\left|x_n-l\right|<\epsilon$ holds.

In this case, we write: $\underset{x\to \infty }{\lim }{x}_n=l\text{ or, }{x}_n\to l\text{ as }n\to \infty .$

From the above Definition1we can say that ,the given sequence $\left\{a_n\right\}$ is a decreasing and bounded below .

From the above Theorem1 we can say that ,the given sequence $\left\{a_n\right\}$ is convergent .

From the above Definition2 we can say that ,the given sequence $\left\{a_n\right\}$ has a limit.

From the above Definition3 we can say about the value of the limit of the given sequence $\left\{a_n\right\}$ .