# real analysis – Is a subset contained in a union of its superset?

For context, I’m taking an introductory real analysis course, and our current topic is intervals. One of the questions requires to prove that, for every $$x, y$$ in some real interval $$I$$, with $$x, the interval $$(x,y)$$ is also contained in $$I$$.

I’m taking a shortcut by proving it’s a subset of an open interval $$I = (a,b)$$, then arguing that $$(x,y) subset (I cup G)$$, for some arbitrary set $$G subset mathbb R$$. From there, changing $$G$$ should span all the cases I’m looking for.

My question: is the provided argument sufficient? Is it a non-trivial fact in set theory? Or am I wrongly using my thesis to prove the hypothesis? Thanks in advance!