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  1. MHB Master
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    #1
    I am reading Manfred Stoll's book: Introduction to Real Analysis.

    I need help with Stoll's proof of Theorem 3.1.16

    Stoll's statement of Theorem 3.1.16 and its proof reads as follows:






    Can someone please help me to demonstrate a formal and rigorous proof of the following:


    If $ \displaystyle U = X \cap O$ for some open subset $ \displaystyle O$ of $ \displaystyle \mathbb{R}$ ...

    ... then ...

    ... the subset $ \displaystyle U$ of $ \displaystyle X$ is open in $ \displaystyle X$ ...



    Help will be much appreciated ...



    My thoughts so far as as follows:

    Suppose $ \displaystyle U = X \cap O$ for some open subset $ \displaystyle O$ of $ \displaystyle \mathbb{R}$ ...


    Need to show $ \displaystyle U$ is open in $ \displaystyle X$ ... that is for every $ \displaystyle p \in U$ there exists $ \displaystyle \epsilon \gt 0$ such that $ \displaystyle N_{ \epsilon } (p) \cap X \subset U$ ... ...


    Now ... let $ \displaystyle p \in U$ ...

    then $ \displaystyle p \in O$ ...

    Therefore there exists $ \displaystyle \epsilon \gt 0$ such that $ \displaystyle N_{ \epsilon } (p) \subset O$ ... since $ \displaystyle O$ is open ...


    BUT ...

    ... how do I proceed from here ... ?




    Hope someone can help ...

    Peter

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  3. MHB Craftsman
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    #2
    Hi Peter,

    Everything looks good so far. From here, what can be said about $N_{\epsilon}(p)\cap X$?

  4. MHB Master
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    #3 Thread Author
    Quote Originally Posted by GJA View Post
    Hi Peter,

    Everything looks good so far. From here, what can be said about $N_{\epsilon}(p)\cap X$?


    Hi GJA ...

    Still perplexed ... can you help further...

    Peter

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    #4
    Hi Peter,

    Think about trying to use $N_{\epsilon}(p)\subset O$ and use that fact to get a set "inequality" for $N_{\epsilon}(p)\cap X$.

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    #5 Thread Author
    Quote Originally Posted by GJA View Post
    Hi Peter,

    Think about trying to use $N_{\epsilon}(p)\subset O$ and use that fact to get a set "inequality" for $N_{\epsilon}(p)\cap X$.



    Thanks GJA ...

    I think the argument you're suggesting is as follows:

    We have $N_{\epsilon}(p)\subset O$

    So therefore $ \displaystyle N_{\epsilon}(p) \cap X \subset O \cap X$ ...

    ... that is $ \displaystyle N_{\epsilon}(p) \cap X \subset U$ ... as required ...


    Is that correct?

    Peter

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    #6
    Yes, this is correct. Nicely done.

  8. MHB Master
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    #7 Thread Author
    Quote Originally Posted by GJA View Post
    Yes, this is correct. Nicely done.

    Thanks for all your help, GJA ...

    It is much appreciated...

    Peter

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