={\color{blue}\frac{\sqrt{k+1}\cdot\sqrt{k+2}+1}{\sqrt{k+2}}>\frac{\sqrt{k+1}\cdot\sqrt{k+{\color{red}1}}+1}{\sqrt{k+2}}=\frac{k+1+1}{\sqrt{k+2}}}=\frac{k+2}{\sqrt{k+2}}=\sqrt{k+2}