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 $\left\{m_k\right\}$and $\left\{M_k\right\}$are sequences that both converge to $L$, and if $m_k\le a_k\le M_k$for all $k$, then $a_k\to L$.

Short answer

The theorem is proved.

Step-by-step solution

Step 1. Given Information.

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

Step 2. Forming the equation.

The sequence $\left\{m_k\right\}$is convergent and converges to $L$.

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

$$\begin{gathered} \left|m_k-L\right|<\epsilon ,k\ge N \\ L-\epsilon <m_k<\epsilon +Lfork\ge N..........\left(1\right) \end{gathered}$$

The sequence $\left\{M_k\right\}$is convergent and converges to $L$.

For given $\epsilon >0$, there exists a positive integer $M$such that

$$\begin{gathered} \left|M_k-L\right|<\epsilon ,k\ge M \\ L-\epsilon <M_k<\epsilon +Lfork\ge M...........\left(2\right) \end{gathered}$$

Step 3. Proving the theorem.

It is given that $m_{k\le }{a}_k\le M_k$

Choose $P=max(N,M)$

Using equation(1) and (2) we get,

$$\begin{gathered} L-\epsilon <m_k\le a_k\le M_k<L+\epsilon fork\ge P \\ L-\epsilon <a_k<L+\epsilon fork\ge P \\ \left|a_k-L\right|<\epsilon fork\ge P \end{gathered}$$

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

$\left|a_k-L\right|<\epsilon fork\ge P$.

Therefore, the sequence$\left\{a_k\right\}$converges to limit$L$.