X^2 = 2 has solution in ℝ

  • Thread starter Bachelier
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  • #1
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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?

Thanks.
 
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  • #2
lavinia
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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?

Thanks.

This isn't true. The square of 1 is not 2
 
  • #3
501
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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.
 
  • #4
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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?

Thanks.

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.
 
  • #5
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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.

Indeed. Please see the attached file for the complete proof.
 

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