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Thread: [SOLVED] real analysis - absolute value

  1. #1
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    [SOLVED] real analysis - absolute value

    If x, y, and are elements of R and x <=z, show that x <= y <= z if and only if |x - y| + |y - z| = |x - z|.
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  2. #2
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    Re: [SOLVED] real analysis - absolute value

    $\displaystyle \textbf{Proof.}$

    $\displaystyle (\Rightarrow) \text{We have that } x \leq y \leq z \text{.}$

    $\displaystyle \text{i) Supose } \left | x - y \right | + \left | y - z \right | < \left | x - z \right | \text{.}$

    $\displaystyle \Rightarrow y - x + z - y < z - x$

    $\displaystyle \Rightarrow z - x < z - x$

    $\displaystyle \Rightarrow \text{Contradiction!}$

    $\displaystyle \text{ii) Supose } \left | x - y \right | + \left | y - z \right | > \left | x - z \right | \text{.}$

    $\displaystyle \Rightarrow y - x + z - y > z - x$

    $\displaystyle \Rightarrow z - x > z - x$

    $\displaystyle \Rightarrow \text{Contradiction!}$

    $\displaystyle \text{Hence, } \left | x - y \right | + \left | y - z \right | = \left | x - z \right | \text{.}$


    $\displaystyle (\Leftarrow) \text{We have that } \left | x - y \right | + \left | y - z \right | = \left | x - z \right | \text{.}$

    $\displaystyle \text{i) Supose } y < x \leq z \text{.}$

    $\displaystyle \Rightarrow x - y + z - y = z - x$

    $\displaystyle \Rightarrow 2x = 2y$

    $\displaystyle \Rightarrow x = y$

    $\displaystyle \Rightarrow \text{Contradiction!}$

    $\displaystyle \text{ii) Supose } x \leq z < y \text{.}$

    $\displaystyle \Rightarrow y - x + y - z = z - x$

    $\displaystyle \Rightarrow 2y = 2z $

    $\displaystyle \Rightarrow y = z $

    $\displaystyle \Rightarrow \text{Contradiction!}$

    $\displaystyle \text{Hence, } x \leq y \leq z \text{.}$
    $\displaystyle \text{Q.E.D.}$
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