Chapter 8.1, Problem 7MC

a. To prove that $\frac{2}{\sqrt{3}}<\frac{3}{\sqrt{5}}$ a student wrote the following:

$\begin{array}{rllll}\frac{3}{\sqrt{5}}-\frac{2}{\sqrt{3}}& =& \frac{3\sqrt{3}-2\sqrt{5}}{\sqrt{15}}& =& \frac{\sqrt{9}\sqrt{3}-\sqrt{4}\sqrt{5}}{\sqrt{15}}\\ & =& \frac{\sqrt{27}-\sqrt{20}}{\sqrt{15}}>0\end{array}$

Explain why this proves that $\frac{2}{\sqrt{3}}<\frac{3}{\sqrt{5}}$.

b. Prove that $\frac{5}{\sqrt{6}}<\frac{6}{\sqrt{7}}$, using the approach in part (a).