Question

Let $\left\{a_k\right\}$be a sequence. Prove the indicated limit rules from Theorem 7.12. You may wish to model your proofs on the proofs of the analogous statements from Section 1.5.

Prove that if $\left|a_k\right|\to 0$, then $a_k\to 0$.

Short answer

The theorem is thus proved.

Step-by-step solution

Step 1. Given Information.

The objective is to prove that$a_k\to 0$.

Step 2. Assumption

The sequence $\left\{a_k\right\}$is convergent and converges to $0$.

By definition of convergence, for $\epsilon >0$there is a positive integer $N$, such that

$$\begin{gathered} \left|\left|a_k\right|-0\right|<\epsilon fork\ge N \\ \left|\left|a_k\right|\right|<\epsilon \\ \left|a_k\right|<\epsilon (Because\left|a_k\right|\ge 0) \end{gathered}$$

Step 3. Proving the theorem.

For $\epsilon >0$, there is a positive integer $N$such that,

$$\begin{gathered} \left|a_k\right|<\epsilon fork\ge N \\ \left|a_k-0\right|<\epsilon fork\ge N \end{gathered}$$

Thus, a positive integer $N$can be found such that $\left|a_k-0\right|<\epsilon$.

Therefore,$\underset{k\to \infty }{\lim }{a}_k=0$