\[\begin{aligned} \frac1{n^\frac12}&=\frac{2}{\sqrt n+\sqrt n}\\ &<\frac{2}{\sqrt{n-1}+\sqrt n}\\ &=2(-\sqrt{n-1}+\sqrt n) \end{aligned}\] \[\begin{aligned} \frac1{n^\frac32}&<\frac{\sqrt n}{\sqrt n\sqrt n\sqrt{n+1}\sqrt{n-1}}\\ &=\frac{\frac{\sqrt{n+1} \sqrt{n}-\sqrt{n} \sqrt{n-1}}{\sqrt{n+1}-\sqrt{n-1}}}{\sqrt n\sqrt n\sqrt{n+1}\sqrt{n-1}}\\ &=(\frac1{\sqrt n\sqrt{n-1}}-\frac1{\sqrt n\sqrt{n+1}})\frac1{\sqrt{n+1}-\sqrt{n-1}}\\ &=(\frac1{\sqrt{n-1}}-\frac1{\sqrt{n+1}})(\frac{\sqrt{n+1}+\sqrt{n-1}}{2\sqrt n})\\ &<\frac1{\sqrt{n-1}}-\frac1{\sqrt{n+1}} \end{aligned}\] \[\begin{aligned} \frac1{\binom nm}-\frac1{\binom{n+1}m}&=\frac{\binom{n+1}m-\binom nm}{\binom nm\binom{n+1}m}\\ &=\frac{\binom n{m-1}}{\binom nm\binom{n+1}m}\\ &=\frac{m}{(n+1)\binom nm}\\ &=\frac{m}{(m+1)\binom{n+1}{m+1}}\\ \end{aligned}\]