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

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

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

Let $S_0 = \O$.

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:
 * $\ds S = \bigcup_{n \mathop = 0}^\infty \paren {S_{n + 1} \setminus S_n}$

From Intersection Distributes over Union:
 * $\ds A \cap S = A \cap \bigcup_{n \mathop = 0}^\infty \paren {S_{n + 1} \setminus S_n} = \bigcup_{n \mathop = 0}^\infty \paren {A \cap \paren {S_{n + 1} \setminus S_n} }$

Therefore: