Q. 67

Let c be a constant, and let $a k$ and $b k$ be convergent sequences with $a k$ as L and $b k$ as

as k.

Verified

Answer

The value is $\lim_{k \to \infty} c a_{k} = c L .$

1Step 1: Given information

The convergence sequence is $\{a_{k}\} such that a_{k} \to L$ .

2Step 2: Calculation.

The goal is to demonstrate $c a_{k} \to c L .$

To demonstrate $c a_{k} \to c L$ , use the sequence's notion of convergence $\{a_{k}\}$ the series $\{a_{k}\}$ converges to L and is convergent.

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

$|a_{k} - L| < \frac{ε}{c} f o r k \geq N$

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

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

For $c \neq 0$ , the value of $|c a_{k} - c L|$ is

\begin{array}{rcl} |c a_{k} - c L| & = & | c | |a_{k} - L| < | c | \frac{ε}{| c |} (Use of 1) \\ = & ε \end{array}

Thus, for $k \geq N, |c a_{k} - c L| < ε$ and hence, $\{c a_{k}\}$ is convergent.