Question

Prove that if $\left\{a_k\right\}$is a sequence of nonzero terms with the property that $\underset{k\to \infty }{\lim }{a}_k=\infty$, then $\frac{1}{a_k}\to 0$.

Short answer

The theorem has been proved.

Step-by-step solution

Step 1. Given Information.

The objective is to show that$\frac{1}{a_k}\to 0$.

Step 2. Proving the theorem.

It is given that $\underset{k\to \infty }{\lim }{a}_k=\infty$, therefore, for given $\epsilon >0$, there exists a positive integer $N$such that

$\left|a_k\right|>\frac{1}{\epsilon }fork>N.........\left(1\right)$

For $k>N.\left|a_k\right|>\frac{1}{\epsilon }$can be written as,

$\frac{1}{\left|a_k\right|}<\epsilon$

Thus, for given, $\epsilon >0$, there exists a positive integer $N$such that

$\frac{1}{\left|a_k\right|}<\epsilon$

Thus,$\frac{1}{a_k}\to 0$.