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

  • Is there a part missing to the question? – vadim123 Apr 29 '13 at 23:35
  • oops, I think you want to clarify your problem statement in your post...if $x, y > 0?$...and what is it to prove? – amWhy Apr 29 '13 at 23:36
  • Have you tried proving the contrapositive? – wj32 Apr 29 '13 at 23:55
up vote 15 down vote accepted

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

  • 2
    This idea popped into my mind immediately after reading the question. – ncmathsadist Apr 30 '13 at 0:15

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 $y = b^2$.

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

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