Q. 72 - Calculus

Question

Prove that if $\left\{a_k\right\}\to L$ and $f$ is a function that is continuous at $L$, then $f\left(a_k\right)\to f\left(L\right)$. 

Step-by-Step Solution

Answer

It is proved that if $\left\{a_k\right\}\to L$ and $f$ is a function that is continuous at $L$, then $f\left(a_k\right)\to f\left(L\right)$. 

Step 1. Given Information

The objective is to prove that if $\left\{a_k\right\}\to L$ and $f$ is a function that is continuous at $L$, then $f\left(a_k\right)\to f\left(L\right)$. 

Step 2. Prove

As $f$ is a continuous function so for a $\epsilon >0$ there exist $\delta >0$ such as $\left|f\left(x\right)-f\left(L\right)\right|<\epsilon \forall x\in N\left(L,\delta \right)$.

Now, $\underset{k\to \infty }{\lim }{a}_k=L$ there exists a $m\in N$ such that

$\left|a_n-L\right|<\delta \forall n\ge m$.

So, for all $n\ge m$, $a_n\in N(L,\delta )$.

So, $\left|f\left(a_n\right)-f\left(L\right)\right|<\epsilon$ for all $n\ge m$.

Hence it is proved that $f\left(a_k\right)\to f\left(L\right)$.