Question

Prove that if $\underset{k\to \infty }{\lim }{a}_k=L,$then localid="1649337757642" $\underset{k\to \infty }{\lim }{a}_{k+1}=L$

Short answer

Hence proved that$\underset{k\to \infty }{\lim }{a}_{k+1}=L$

Step-by-step solution

Step 1. Given information

The given sequence$\underset{k\to \infty }{\lim }{a}_k=L$

Step 2. The strategy to prove that limk→∞ak+1=L.

Use the defination of convergence of sequence$\left\{a_k\right\}$

The sequence$\left\{a_k\right\}$ is convergent and convergers to L.

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

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

The result $\left|a_k-L\right|<\epsilon$is true for all $k\ge N$

Since the following inequality holds

$k+1>k$

Therefore

$k+1>N(Becausek\ge N)$

Therefore,there is a positive integer Nsuch that

$\left|a_{k+1}-L\right|<\epsilon fork+1>N$

Thus,$\underset{k\to \infty }{\lim }{a}_{k+1}=L$