How to prove that outer measure $|A|=\lim_{t\to \infty}|A\cap (-t,t)|$?

Let $A\subset R$. Then, $|A|=\lim_{t\to \infty}|A\cap (-t,t)|\tag 1$

$|A|:=\inf\{\sum_j l(I_j): A\subset \cup I_j\},$ where $I_j$'s are open intervals in $\mathbb R$ and $l(I_j)$ is the length of the interval $I_j$, and infimum is taken over all sequences of open intervals whose union contains $A$.

I am trying to prove $(1)$.

I know that $|A|= |A\cap (-t,t)|+|A\cap (-t,t)^c|, t>0$.

Suppose that $|A|<\infty$. Then, it suffices to prove that $\lim_{t\to\infty}|A\cap(-t,t)^c|=0$.

Define $f(t):=|A\cap (-t,t)^c|,t>0$. Then, $f$ is a monotonically decreasing function. So $\lim_{t\to \infty}f(t)= \inf_{t>0} f(t)=p,$ say.

If it is shown that $p=0$, then the proof is complete. Suppose that $p>0$. I don't understand how to get a contradiction from here.

And if $|A|=\infty$, then I don't know how to proceed.

Please help. Thanks.