Question

Let c be a constant, and let $ak$and $bk$be convergent sequences with $ak$as L and $bk$as

as k.

Short answer

The value is$\underset{k\to \infty }{\lim }c{a}_k=cL.$

Step-by-step solution

Given information

The convergence sequence is$\left\{a_k\right\}\mathrm{such}\mathrm{that}{a}_k\to L$.

Calculation.

The goal is to demonstrate $c{a}_k\to cL.$

To demonstrate $c{a}_k\to cL$, use the sequence's notion of convergence$\left\{a_k\right\}$the series$\left\{a_k\right\}$converges to L and is convergent.

When given $\epsilon >0$, there is an integer N that is positive and such that,

$\left|a_k-L\right|<\frac{\epsilon }{c}fork\ge N$

Each term in the sequence is $0$when $c=0$.

The series stabilizes and eventually converges to zero, becoming a constant sequence.

For $c\ne 0$, the value of $\left|c{a}_k-cL\right|$is

localid="1661332238088" $\begin{array}{rcl}\left|c{a}_k-cL\right|& =& \left|c\right|\left|a_k-L\right|\left.<\left|c\right|\frac{\epsilon }{\left|c\right|}\text{(Use of}1\right)\\ & =& \epsilon \end{array}$

Thus, for $k\ge N,\left|c{a}_k-cL\right|<\epsilon$and hence,$\left\{c{a}_k\right\}$is convergent.