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The following is my book's proof that $\sup\left\{x\in\mathbb{Q}:x>0, ~ x^2<2\right\} = \sqrt{2}.$
I don't follow the bit where it says "if [tex]s[/tex] were irrational, then [tex]w = \frac{\lfloor(n+1)s\rfloor}{n+1}+\frac{1}{n+1}.[/tex]"
Could someone please elaborate on that bit? How does s being irrational imply that?
I don't follow the bit where it says "if [tex]s[/tex] were irrational, then [tex]w = \frac{\lfloor(n+1)s\rfloor}{n+1}+\frac{1}{n+1}.[/tex]"
Could someone please elaborate on that bit? How does s being irrational imply that?
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