Answer with Step-by-step explanation:
We are given that
[tex]A\subseteq R[/tex]
Let f:[tex]A\rightarrow R[/tex]
[tex]c\in R[/tex] be a cluster point of A.
[tex]\lim_{(x\rightarrow c)}f(x)[/tex] exist let
[tex]\lim_{(x\rightarrow c)}f(x)=L[/tex]
If [tex]\mid f\mid[/tex] denotes the function and
[tex]\mid f(x)\mid=\mid f\mid (x)[/tex] for [tex]x\in A[/tex]
We have to prove that [tex]\mid \lim_(x\rightarrow c)f(x)\mid=\lim_(x\rightarrow c)\mid f\mid (x)[/tex]
[tex]\mid \lim_{(x\rightarrow c)}f(x)\mid=\lim_{(x\rightarrow c)}\mid f(x)\mid[/tex]
Substitute the value then we get
[tex]\mid \lim_{(x\rightarrow c)}f(x)\mid=\mid L\mid =\lim_{(x\rightarrow c)}\mid f\mid (x)[/tex]
Hence proved.
