Question

Fill in the blanks to complete each of the following theorem statements.

Basic Limit Rules for Convergent Sequences: If $\left\{a_k\right\}\mathrm{and}\left\{b_k\right\}\mathrm{are}\mathrm{convergent}\mathrm{sequences}\mathrm{with}{a}_k\to L\mathrm{and}{b}_k\to M\mathrm{as}k\to \infty$ and if c is any constant,

If {ak} is both bounded and (eventually) monotonic, then {ak}______ .

Short answer

The required answer is {ak} converges.

Step-by-step solution

Step 1. Given Information    

The given data is if $\left\{a_k\right\}\mathrm{and}\left\{b_k\right\}\mathrm{are}\mathrm{convergent}\mathrm{sequences}\mathrm{with}{a}_k\to L\mathrm{and}{b}_k\to M\mathrm{as}k\to \infty$

Step 2. Explanation        

Using the theorem of bounded monotonic sequence always converge.

If {ak} is both bounded and (eventually) monotonic, then {ak}converges.