请求验证Apostol著作中R¹非空开集构成区间定理的证明是否正确

Hey everyone! I'm currently working my way through Apostol's text, and all the definitions and theorems below are taken directly from his book:

Definition (Component Interval) #

Let $S$ be an open subset of $\mathbb{R}^1$. An open interval $I$ (which can be finite or infinite) is called a component interval of $S$ if:

• $I \subseteq S$, and
• There does not exist an open interval $J$ distinct from $I$ such that $I \subseteq J \subseteq S$.

Theorem #

Every point in a non-empty open set $S$ belongs to exactly one component interval of $S$.

I've finished writing up my proof for this theorem, and I'd really appreciate it if any of you could take a look and verify whether my entire proof process is correct. Thanks so much in advance!

内容的提问来源于stack exchange，提问作者user370967