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Math Help - Interval Proof

  1. #1
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    Interval Proof

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

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

     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  (a, \infty) \subseteq [b, \infty) \Leftrightarrow a \geq b .
    Forward.
    If x\in (a,\infty) then x\in [b,\infty) thus, a<x \implies b\leq x.
    Argue by contradiction, say a<b then choosing x=(a+b)/2. This leads to contradiction, why?

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