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Thread: Analysis - Compactness

  1. #1
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    Analysis - Compactness

    Hello!

    I am trying to prove that K is compact (from the definition of compactness), where $\displaystyle K \subset R$ is defined by $\displaystyle K = \{\frac{1}{n} : n \in N\} \cup \{0\}$.

    Any help is greatly appriciated!
    Thanks!
    Matt
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    Quote Originally Posted by matt.qmar View Post
    Hello!

    I am trying to prove that K is compact (from the definition of compactness), where $\displaystyle K \subset R$ is defined by $\displaystyle K = \{\frac{1}{n} : n \in N\} \cup \{0\}$
    If $\displaystyle \varepsilon > 0$ then the open interval $\displaystyle \left( {0,\varepsilon } \right)$ contain almost all the points of $\displaystyle K$.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by matt.qmar View Post
    Hello!

    I am trying to prove that K is compact (from the definition of compactness), where $\displaystyle K \subset R$ is defined by $\displaystyle K = \{\frac{1}{n} : n \in N\} \cup \{0\}$.

    Any help is greatly appriciated!
    Thanks!
    Matt
    It may be more clear (at least to me) that in any metric space, if $\displaystyle x_n\to x$ then $\displaystyle \{x\}\cup\{x_n:n\in\mathbb{N}\}$ is compact. Use Plato's idea of looking at a neighborhood of the convergence point.
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