<s>[INST] which is larger number? $\sqrt{7}-\sqrt{6}$ or $\sqrt{6}-\sqrt{5}$ Which is larger number? $\sqrt{7}-\sqrt{6}$ or $\sqrt{6}-\sqrt{5}$?
Squaring both sides will give me something but I could not go any further.
 [/INST] You can also use the inequality between the arithmetic mean and the quadratic mean: $$ \frac{x+y}{2} \leq \sqrt{\frac{x^2 + y^2}{2}}. $$ Setting $x = \sqrt{5}$ and $y = \sqrt{7}$ gives you $$\frac{\sqrt{5} + \sqrt{7}}{2} \leq \sqrt{6},$$ which is equivalent to $$ \sqrt{7} - \sqrt{6} \leq \sqrt{6} - \sqrt{5}.$$
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