Chapter 8, Problem 16RQ

To determine

To find: The statement is true or false and if the statement then given the explanation.

Answer to Problem 16RQ

Thestatement is false as $\langle a_n+b_n\rangle =\langle 0,0,0,,,,,0\rangle$ .

Explanation of Solution

Given:

Thegiven statement is that if $\left\{a_n\right\}$ is decreasing and $a_n>0$ for all values of $n$ , then $\left\{a_n\right\}$ is convergent.

Calculation:

The given statement is that if $\left\{a_n\right\}$ is decreasing and $a_n>0$ for all values of $n$ , then $\left\{a_n\right\}$ is convergent.

From the statement it is concluded that the sequence is monotonic and bounded above, thus the conclusion is that the sequence is convergent by the convergent monotonic sequence theorem.

Thus, the statement is true by the convergent monotonic sequence theorem.