Outer Measure of Limit of Increasing Sequence of Sets/Proof 1

Proof
By the monotonicity of $\mu^*$, it suffices to prove that:
 * $\displaystyle \mu^* \left({A \cap S}\right) \le \lim_{n \mathop \to \infty} \mu^* \left({A \cap S_n}\right)$

Assume that $\mu^* \left({A \cap S_n}\right)$ is finite for all $n \in \N$, otherwise the statement is trivial by the monotonicity of $\mu^*$.

Let $S_0 = \varnothing$.

Then $x \in S$ there exists an integer $n \ge 0$ such that $x \in S_{n + 1}$.

Taking the least possible $n$, it follows that $x \notin S_n$, and so:
 * $x \in S_{n + 1} \setminus S_n$

Therefore:
 * $\displaystyle S = \bigcup_{n \mathop = 0}^\infty \left({S_{n + 1} \setminus S_n}\right)$

From Intersection Distributes over Union:
 * $\displaystyle A \cap S = A \cap \bigcup_{n \mathop = 0}^\infty \left({S_{n + 1} \setminus S_n}\right) = \bigcup_{n \mathop = 0}^\infty \left({A \cap \left({S_{n + 1} \setminus S_n}\right)}\right)$

Therefore: