# Thread: Union and Intersection of Neighborhoods

1. ## Union and Intersection of Neighborhoods

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

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

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

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

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

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