$\varepsilon$-neighborhood set notation as $V_{\varepsilon}(a)$ and proof question Two questions regarding the set $V_{\varepsilon}(a)$.  It is defined in the book I am currently using as
$$V_{\varepsilon}(a)=\{x\in\mathbb{R}:|x-a|<\varepsilon\}$$
1).  Why the $V$?  I suppose there is a reason for using most notation and abbreviation (like using $\mathbb{R}$ for the reals, or $n$ to represent a natural number...)
2).  Let $\varepsilon>0, \delta>0, a\in\mathbb{R}$.  Show that $V_{\varepsilon}(a)\cap V_{\delta}(a)$ and $V_{\varepsilon}(a)\cup V_{\delta}(a)$ are $\gamma$-neighborhoods of $a$ for appropriate values of $\gamma$.
So I say, 
$$V_{\varepsilon}(a)\cap V_{\delta}(a)=\{x\in\mathbb{R}:|x-a|<\varepsilon \text{ and } |x-a|<\delta\}$$
$$=\{x\in\mathbb{R}:|x-a|<\min{(\varepsilon,\delta)}\} $$
Letting $\gamma=\min{(\varepsilon,\delta)},$
$$=\{x\in\mathbb{R}:|x-a|<\gamma\}=V_{\gamma}(a)$$
Therefore, $V_{\varepsilon}(a)\cap V_{\delta}(a)$ is a $\gamma$-neighborhood of $a$.  
If this is all i need, I assume the union of two neighborhoods is also a $\gamma$-neighborhood by using the $\max$ function instead?
 A: For your first question, I can only wager an educated guess, but it's likely that $V$ comes from the French voisinage.  I should note that I don't speak or read French, but according to google voisinage translates generally as "vicinity" or "neighborhood," and which seems to be the proper translation of the "neighborhood" in a mathematical context, based on a search of MathSciNet and Wikipedia (English and French).
Since a decent amount of work in analysis and topology was published in German or French, it's not uncommon to see variables takes from words in these languages.  Some other examples that come to mind are $G_\delta$, from the German Gebeit and Durchschnitt ("open" and "intersection" repectively), $F_\sigma$ from the French fermé and somme ("closed" and "union"), and $K$ or $\mathbb{K}$ for a field, from the German Korpus.
As far as your proof is concerned, it's certainly correct.   You have the right idea for unions, too.  Note that depending on the definition, you might be proving more than is strictly necessary.  In particular, it depends on precisely what an $\varepsilon$-neighborhood is.  The most natural definition to me is that $U$ is a $\varepsilon$-neighborhood of $a \in \mathbb{R}$ if and only if $U$ contains some $V_\varepsilon(a)$, i.e., if and only if there exists $\varepsilon > 0$ such that $V_\varepsilon(a) \subseteq U$.  In this case, you've proven that $V_\varepsilon(a) \cap V_\delta(a)$ is in fact a set of the form $V_\gamma(a)$, so it's certainly a $\gamma$-neighborhood of $a$.
