$A$ is Lebesgue measurable iff $\forall \epsilon > 0$, there exists open set $G$ such that $|G \setminus A| + |A \setminus G| < \epsilon$

Question

In Sheldon Axler's Measure Theory book, I came across this problem -

Suppose $A \subset \mathbb{R}$ and $|A| < \infty$. Prove that $A$ is Lebesgue measurable if and only if for every $\epsilon > 0$ there exists a set $G$ that is the union of finitely many disjoint bounded open intervals such that $|A \setminus G| + |G \setminus A| < \epsilon$.

Here, $|A|$ stands for the outer measure of $A$.

The definition of Lebesgue measurable sets are as follow -

$A$ is Lebesgue measurable iff -

1. For each $\epsilon > 0$, there exists a closed set $F \subset A$ with $|A \setminus F| < \epsilon$.

2. There exists a Borel set $B \subset A$ such that $|A \setminus B| = 0$.

3. For each $\epsilon > 0$, there exists an open set $G \supset A$ such that $|G \setminus A| < \epsilon$

4. There exists a Borel set $B \supset A$ such that $|B \setminus A| = 0$

I have proved the $\implies$ part, but I am stuck in the converse part. Any help is appreciated.