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Math Help - Real analysis help please

  1. #1
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    Question Real analysis help please

    I'm having a lot of trouble with contractions and open coverings for my real analysis class, and I was hoping someone would be able to help me with a few of the problems:

    1) Give an example of a continuous bounded function on (-inf, inf) that attains neither a maximum nor minimum value

    2a) For all x in (0,1), let I_x denote the open interval (x/2, (x+1)/2). Show that the family G of all such I_x is an open covering of (0,1) which admits no finite subcovering of (0,1)
    2b) Add two appropriate sets to the family G (from above) to form an open covering H of [0,1]. Show that H does admit a finite subcovering of [0,1].
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  2. #2
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    Quote Originally Posted by Natedogg5106 View Post
    1) Give an example of a continuous bounded function on (-inf, inf) that attains neither a maximum nor minimum value
    Do you know about \arctan(x)?
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Natedogg5106 View Post
    2a) For all x in (0,1), let I_x denote the open interval (x/2, (x+1)/2). Show that the family G of all such I_x is an open covering of (0,1) which admits no finite subcovering of (0,1)
    2b) Add two appropriate sets to the family G (from above) to form an open covering H of [0,1]. Show that H does admit a finite subcovering of [0,1].
    2a) So you are attempting to show that \mathcal{I}=\left\{I_x=\left(\frac{x}{2},\frac{x+1  }{2}\right):x\in(0,1)\right\} is an open cover of (0,1). Did you try it? Did you show that (0,1)\subseteq \bigcup_{I_x\in\mathcal{I}}I_x? What happens if you take one of the elmements out?

    2b). Hint: When you add the two appropriate sets you only need three intervals to cover it.
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    Alright thank you very much. I didn't even think to use any inverse trigonometric functions. I think I figured out the second problem, although the two sets I added probably aren't the right ones, it seems to work to me. How do I give you guys feedback or whatnot? This is my first time using this forum
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Natedogg5106 View Post
    Alright thank you very much. I didn't even think to use any inverse trigonometric functions. I think I figured out the second problem, although the two sets I added probably aren't the right ones, it seems to work to me. How do I give you guys feedback or whatnot? This is my first time using this forum
    Either press the thank button, which you did, or press the gree plus box where it says "rep".
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