I want to check if my reasoning and mathematical language is correct here. There's only two cases to check for. One where one of the neighborhoods is inside the other, and the other where the two neighborhoods overlap without enclosing each other.
Definitions being used: $\epsilon$-neighborhood: let $a \in \mathbb{R}$ and $\epsilon > 0$. Then $(a-\epsilon,a+\epsilon)$ is called an $\epsilon$-neighborhood and is denoted $J_{\epsilon}(a)$.
-Case 1: say $U = (a,b), V = (c,d)$ where $a \le c < d \le b$.
In terms of $\epsilon$-neighborhoods: $U = J_{\delta}(a^\prime)$ where $a^{\prime} = \frac{a+b}{2}$ and $V = J_{\epsilon}(c^\prime)$ where $c^{\prime} = \frac{c+d}{2}$.
Now to show $U \cup V$ is also a $\epsilon$-neighborhood: $U \cup V = (a,b) = U = J_{\delta}(a^{\prime})$.
Now to show $U \cap V$ is also a $\epsilon$-neighborhood: $U \cap V = (c,d) = V = J_{\epsilon}(c^{\prime})$
-Case 2: say $U = (a,b), V = (c,d)$ where $a < c < b < d$.
In terms of $\epsilon$-neighborhoods: $U = J_{\delta}(a^\prime)$ where $a^{\prime} = \frac{a+b}{2}$ and $V = J_{\epsilon}(c^\prime)$ where $c^{\prime} = \frac{c+d}{2}$.
Now to show $U \cup V$ is also a $\epsilon$-neighborhood: $U \cup V = (a,d)$, where $a<d$ and $U \cup V = J_{\beta}(a^{\prime\prime})$, where $a^{\prime\prime} = \frac{a+d}{2}$.
Now to show $U \cap V$ is also a $\epsilon$-neighborhood: $U \cap V = (c,b)$, where $c<b$ and $U \cap V = J_{\alpha}(c^{\prime\prime})$, where $c^{\prime\prime} = \frac{c+b}{2}$.
Is this an ok proof in terms of the mathematical language? Thanks!
Underlying everything you write are the notions that the non-empty intersection of open intervals is an open interval and that an open interval of $\mathbb{R}$ is equivalent to an $\epsilon$-ball about some point. Right now when you check that an interval has a corresponding $\epsilon$-ball, you just calculate the new center of the interval. You might also want to note what the new $\epsilon$ is.
Clarity above all.
The idea of some of these simple proofs in analysis is that you rigorously go through arguments about simple things to get practice making such arguments. You should show, for example, that $U\cap V\subseteq J_{\alpha}(c'')$ and then that $J_{\alpha}(c'')\subseteq U\cap V$ to conclude that the two sets are the same. It's possible that you're not expected to make such complete arguments, but I was.
Also, for the first case, you don't seem to formally use your statement "In terms of $\epsilon$-neighborhoods...". While the statement is illustrative, I'm not convinced that it is strictly necessary.
When you conclude that $U\cup V=U$ and $U\cap V=V$, you're implicitly using the fact that that $V\subseteq U$. For your proof to be clear, you might want to emphasize how you're reaching such conclusions.
There are other things you can spot on your own.