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Thread: Interval Proof

  1. #1
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    Interval Proof

    We want to prove that $\displaystyle (a, \infty) \subseteq [b, \infty) \Leftrightarrow a \geq b $.

    Proof: $\displaystyle x \in (a, \infty) \Rightarrow x \in [b, \infty) \Rightarrow a \geq b $.

    $\displaystyle a \geq b \Rightarrow x \in (a, \infty) \Rightarrow x \in [b, \infty) \Rightarrow (a, \infty) \subseteq [b, \infty) $.

    Does this look right? I drew a number line and used 'intuitive reasoning' to show that it is true.

    Thanks
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  2. #2
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    Quote Originally Posted by shilz222 View Post
    We want to prove that $\displaystyle (a, \infty) \subseteq [b, \infty) \Leftrightarrow a \geq b $.
    Forward.
    If $\displaystyle x\in (a,\infty)$ then $\displaystyle x\in [b,\infty)$ thus, $\displaystyle a<x \implies b\leq x$.
    Argue by contradiction, say $\displaystyle a<b$ then choosing $\displaystyle x=(a+b)/2$. This leads to contradiction, why?

    Converse.
    If $\displaystyle x\in (a,\infty)$ then $\displaystyle a<x$, but then $\displaystyle b\leq a < x$. Thus $\displaystyle x\in [b,\infty)$. Thus $\displaystyle (a,\infty)\subseteq [b,\infty)$.
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  3. #3
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    contradicts the fact that $\displaystyle b \leq \frac{a+b}{2} $.
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