I'm having trouble understanding a proof of "the neighborhoods in R^n are open"

Question

Lemma 137: "The neighborhoods in R^n are open."

Proof

I can follow along and visualize the proof, but how does the final statement prove that each point within the neighborhood of x0 is an interior point, therefore making any neighborhood of R^n open?

Answer 1

By definition: $x$ is an interior point of $S$ if there exists an open ball $B$ containing $x$ such that $B\subseteq S$.

The proof shows that given $x\in B_\varepsilon(x_0)$ one can construct the ball $B_{\varepsilon'}(x)$ around $x$ which is contained in $B_\varepsilon(x_0)$. That is, the set $B_\varepsilon(x_0) \sim S$ and $B_{\varepsilon'}(x)\sim B$ in the above definition. Thus, the proof constructs the open ball $B$ around $x$ which is contained in $S$. This shows that $x$ is an interior point.