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Math Help - easy question

  1. #1
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    easy question

    Let M a metric space and z,x\in M, d a metric on M

    If i have d(z,x)<\epsilon , \forall \epsilon > 0


    is true that z=x ?

    if this is true, how i can prove that?

    thanks
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  2. #2
    Super Member Matt Westwood's Avatar
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    Suppose z \ne x.

    Then \exists \epsilon_1 \in \mathbb{R}^+: d (z, x) = \epsilon_1.

    Then \exists \epsilon \in \mathbb{R}^+: \epsilon = \epsilon_1 / 2: \epsilon < d (z, x).

    So z \ne x \Longrightarrow \exists \epsilon \in \mathbb{R}^+: \epsilon < d (z, x).

    The result follows by transposition.
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  3. #3
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    I answer to my inself

    Suposse that z\neq x

    This implies that d(z,x)>0, d(z,x)=h, h>0

    But for the arquimedean property of IR, \exists n\in \mathbb{N}: 0<\frac{1}{n}<h .This implies that \exists f:=\dfrac{1}{n}  \in \mathbb{R}:d(z,x)>f. Contradiction.

    Then, z=x
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  4. #4
    Super Member Matt Westwood's Avatar
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    Perfect.
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