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 {mk} and {Mk} are convergent sequences that both converge to L, and if$m_k\_\_\_\_a_k\_\_\_\_\_\_M_k\mathrm{for}\mathrm{all}k\mathrm{then}{a}_k\to \_\_\_\_\_$

Short answer

The required answer is$m_k\mathbf{\le }{a}_k\mathbf{\le }{M}_k\mathrm{for}\mathrm{all}k\mathrm{then}{a}_k\to \mathit{L}$

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 concept of squeeze theorem of sequences,

If {mk} and {Mk} are sequences that both converge toL, and if$m_k\mathbf{\le }{a}_k\mathbf{\le }{M}_k\mathrm{for}\mathrm{all}k\mathrm{then}{a}_k\to \mathit{L}$