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Math Help - prove that u is open

  1. #1
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    prove that u is open

    Prove that U \subset M is open if and only if whenever (Xn)^\infty_{n=1} is a sequence of elements of M converging to some x \in U, then there exists N \in N , such that  Xn \in U for every n>=N.

    this is one of the question in my midterm test that i couldnt do, the prof is not giving out solution, need some help to prove this
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: prove that u is open

    Quote Originally Posted by wopashui View Post
    Prove that U \subset M is open if and only if whenever (Xn)^\infty_{n=1} is a sequence of elements of M converging to some x \in U, then there exists N \in N , such that  Xn \in U for every n>=N.

    this is one of the question in my midterm test that i couldnt do, the prof is not giving out solution, need some help to prove this
    So, we're in a metric space, right? Assume first that U is open but that this does not hold. Then, there exists x\in U and a sequence x_n\to x failing to eventually in U. Clearly then any neighborhood x is going to contain all but finitely many of the x_n and so, by assumption, an element of M-U. Thus, there does not exist a neighborhood of x contained in U, which contradicts openess. Conversely, if U is open and x_n\to x\in U you know that there exists a neighborhood V of x with V\subseteq U. But, since x_n\to x we have by definition that all but finitely many x_n\in V\subseteq U.
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