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X^2 = 2 has solution in ℝ

  1. Dec 21, 2012 #1
    In one of my old notes, I was reviewing this proof. It started with:

    Consider [itex] S = \{ x \in \mathbb{R} \ | \ x^2 = 2, \ x > 0\} [/itex], then S is not empty because [itex]1 \in S[/itex]

    Why is 1 in the set?

    Last edited: Dec 21, 2012
  2. jcsd
  3. Dec 21, 2012 #2


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    This isn't true. The square of 1 is not 2
  4. Dec 21, 2012 #3
    x^2 = 2 doesn't seem like a particular interesting thing to look at. I can't help but notice that there are two "2's" in your expression. Perhaps you were looking at a specific case of the set x^n = n. The number 1 would be in this set for the case that n=1.
  5. Dec 21, 2012 #4
    It's probably a typo and it should be

    [tex]S=\{x\in \mathbb{R}~\vert~x^2\leq 2,~x>0\}[/tex]

    I bet they end up taking the supremum of S.
  6. Dec 21, 2012 #5
    Indeed. Please see the attached file for the complete proof.

    Attached Files:

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