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, then

Ifrole="math" localid="1649231795638" $a_k\to L\mathrm{and}{a}_k\to M\mathrm{then}\mathrm{\_\_\_\_\_\_\_}.$

Short answer

The required answer is$L=M$

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 uniqueness of limits for sequences,

If$a_k\to L\mathrm{and}{a}_k\to M\mathrm{then}\mathit{L}\mathbf{=}\mathit{M}$