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I am stuck on this homework problem!

Prove that if $x$ and $y$ are real numbers such that $0<x<y$ , then $\sqrt{x}<\sqrt{y}$.

This is in a chapter involving the least upper bound axiom and the Archimedean Principle, but I cannot figure how to use either of those to prove this! Any help would be most appreciated! Thanks

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  • $\begingroup$ Is there a part missing to the question? $\endgroup$
    – vadim123
    Apr 29, 2013 at 23:35
  • $\begingroup$ oops, I think you want to clarify your problem statement in your post...if $x, y > 0?$...and what is it to prove? $\endgroup$
    – amWhy
    Apr 29, 2013 at 23:36
  • $\begingroup$ Have you tried proving the contrapositive? $\endgroup$
    – wj32
    Apr 29, 2013 at 23:55

3 Answers 3

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Hint : $ y-x=(\sqrt{y}+\sqrt{x})(\sqrt{y}-\sqrt{x})$ Now look at the signs..

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    $\begingroup$ This idea popped into my mind immediately after reading the question. $\endgroup$
    – ncmathsadist
    Apr 30, 2013 at 0:15
  • $\begingroup$ This was really beautiful $\endgroup$
    – Arkilo
    Feb 10, 2020 at 19:01
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Suppose $0 < a < b$, multiplying by a gives us, $0 < \color{brown}{a^2 < ab}$.
Similarly, multiplying by $b$ gives $0 < \color{green}{ab < b^2}$ so $ab < b^2$. So from the brown and green, we have $0 < a < b \Leftrightarrow 0 < a^2 < b^2$.

Notice that your result is a corollary of this, let $x = a^2$ and $y = b^2$.

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Hint: write it in contrapositive form - if $\sqrt{x}\geq \sqrt{y}$, then $x\geq y$ (assuming $x,y>0$).

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