<s>[INST] Arrange $x,y,z$ in ascending order. $c>1, c \in \mathbb{R}$
$x = \frac{\sqrt{c+2} - \sqrt{c+1}}{\sqrt{c} - \sqrt{c-1}}
     =  (\sqrt{c+2} - \sqrt{c+1})(\sqrt{c} + \sqrt{c-1})$
$y = \frac{\sqrt{c+2} - \sqrt{c+1}}{\sqrt{c + 1} - \sqrt{c}}
     = (\sqrt{c+2} - \sqrt{c+1})(\sqrt{c+1} + \sqrt{c})$
$z = \frac{\sqrt{c} - \sqrt{c-1}}{\sqrt{c+2} - \sqrt{c+1}}
     = (\sqrt{c} - \sqrt{c-1})(\sqrt{c+2} + \sqrt{c+1})$
It is pretty easy to prove that $x<y$. I have already done that. By experimental substitution, I have found out that $y<z$. But every time I try out prove this generically, I end up with LHS and RHS both looking terrible and yielding unwanted false results and conclusions like $ \sqrt x < 0$. Can someone please help me out in proving that $ y<z $??? Please. Even a definitive good hint is helpful.
 [/INST] Hint One way to look at it is that $t\mapsto \sqrt t$ is concave, so successive secants have lower slope. Now can you relate $z,x$ by considering them as ratios of slopes?
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