**Definition used:**

A sequence
{an}
has a limit *L*, if every
ε>0
there exists an integer *N* such that
|an−L|<ε for n>N
. That is,
limn→∞an=L
.

**Proof:**

Given that,
limn→∞ an=L
and the function *f* is continuous at *L.*

It is enough to prove that, for given
ε>0
and there is some integer *N* such that
|f(an)−f(L)|<ε
when
n>N
.

Since *f* is continuous at *L* and there is some number
δ>0
such that,

|f(x)−f(L)|<ε if |x−L|<δ
(1)

Since
limn→∞ an=L
and by the definition stated above,
|an−L|<δ if n>N
.

Then,
0<|an−L|<δ
and by the equation (1),
|f(an)−f(L)|<ε
.

That is,
limn→∞ f(an)=f(L)

Hence the required proof is obtained.