# Thread: Union and Intersection of Neighborhoods

1. ## Union and Intersection of Neighborhoods

Let $\epsilon>0, \delta>0$ and $a \in \mathbb{R}$. Show that $V_\epsilon(a) \cap V_\delta(a)$ and $V_\epsilon(a) \cup V_\delta(a)$ are $\gamma$-neighborhoods of $a$ for appropriate values of $\gamma$.

Any hints will be appretiate. I have a proof, but it's really simple and I doubt it's correct. Thanks!

2. Originally Posted by dubito
Let $\epsilon>0, \delta>0$ and $a \in \mathbb{R}$. Show that $V_\epsilon(a) \cap V_\delta(a)$ and $V_\epsilon(a) \cup V_\delta(a)$ are $\gamma$-neighborhoods of $a$ for appropriate values of $\gamma$.

Any hints will be appretiate. I have a proof, but it's really simple and I doubt it's correct. Thanks!
What is a $\gamma$ neighborhood.

3. Don't know that's all the information I was given.

4. Originally Posted by dubito
Let $\epsilon>0, \delta>0$ and $a \in \mathbb{R}$. Show that $V_\epsilon(a) \cap V_\delta(a)$ and $V_\epsilon(a) \cup V_\delta(a)$ are $\gamma$-neighborhoods of $a$ for appropriate values of $\gamma$.
Let $\gamma = \min \left\{ {\delta ,\varepsilon } \right\}\,\& \,\Gamma = \max \left\{ {\delta ,\varepsilon } \right\}$.
Under the usual undestanding of $\gamma-\text{neighborhoods}$,
$V_\epsilon(a) \cap V_\delta(a)=V_{\gamma}(a)$ and $V_\epsilon(a) \cup V_\delta(a)=V_{\Gamma}(a)$.

5. Thanks that's similar to what I had.