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,

Let {ak} be a sequence that converges to L. Then every subsequence of{ak}________.

Short answer

The required answer isevery subsequence of {ak} also converges toL.

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 subsequences of convergent sequences converge.

Let {ak} be a sequence that converges to L. Then every subsequence of {ak} also converges to L.