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Math Help - Union and Intersection of Neighborhoods

  1. #1
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    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!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by dubito View Post
    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.
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  3. #3
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    Don't know that's all the information I was given.
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    Quote Originally Posted by dubito View Post
    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).
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  5. #5
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    Thanks that's similar to what I had.
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