Question

Prove Theorem 7.9. That is, let $a:[1,\infty )\to \mathrm{ℝ}$be a continuous function and let $a_k=a\left(k\right)$for every $k\in {\mathrm{\mathbb{Z}}}^{+}$. Show that if $\underset{x\to \infty }{\lim }a\left(x\right)=L$, then $\left\{a_k\right\}\to L$

Short answer

The theorem is hence proved.

Step-by-step solution

Step 1. Given Information.

The objective is to show that if $\underset{x\to \infty }{\lim }a\left(x\right)=L$then $\left\{a_k\right\}\to L$.

Step 2. Forming the equation.

It is given that $\underset{x\to \infty }{\lim }a\left(x\right)=L$, therefore, for given, $\epsilon >0$, there exists a positive integer such that

$\left|a\left(x\right)-L\right|<\epsilon forx>N.......\left(1\right)$

Also, $a\left(k\right)=a_k$

Therefore, using equation (1) we get,

$$\begin{gathered} \left|a\left(k\right)-L\right|<\epsilon fork>N......\left(2\right) \\ \left|a_k-L\right|<\epsilon fork>N \end{gathered}$$

Step 3. Proving the theorem.

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

$\left|a_k-L\right|<\epsilon fork>N$

Hence,$\left\{a_k\right\}\to L$.