Question

Prove Theorem 7.14. That is, show that if $\left\{a_k\right\}$ is a sequence that converges to L, then every subsequence of$\left\{a_k\right\}$ also converges to L

Short answer

Proved that every subsequence of the sequence $\left\{a_k\right\}$converges to the same limit L

Step-by-step solution

Step 1. Given information

The given sequence $\left\{a_k\right\}$ converging to limit L

Step 2. Finding the subsequence of ak

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

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

Since $\left\{a_k\right\}$is a subsequence of sequence $\left\{a_k\right\}$; therefore

$n_k\ge k$

The inequalities $k\ge Nand{n}_k\ge kimplies{n}_k\ge k$

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

$\left|a_{n_k}-L\right|<\epsilon for{n}_k\ge k$

Thus,subsequence $\left\{a_{n_k}\right\}$ converges to L

Therefore,every subsequence of the sequence$\left\{a_k\right\}$ converges to the same limitL