I'm new in here. Considering my person: I am physics student (BSc.) who has finished 2 semesters by now. Within the first two semesters, I discovered that mathematics is beautiful and that I want to learn maths in more depth than the courses at my university teach it. Therefore I'm trying to self-study the book "Principles of Mathematical Analysis", by Walter Rudin, in my leisure time. I have a question concerning
Chapter 2, Exercise 1: Prove that the empty set is a subset of every set.
My first intention would have been:
Let X be an arbitrary set, then $\varnothing \cup X = X \cup \varnothing = X \implies \varnothing \subseteq X$ since $S \subseteq S \cup T$ and $T \subseteq S \cup T$ for arbitrary sets $S$ and $T$. $X$ was arbitrary, thus the assertion holds. $\Box$
But I'm not completely sure, if the prove is complete, as I haven't seen this version anywhere within a quick google search. What would you say?