Square root Taylor series

Question

I want to verify the inequality: $$ \sqrt{n+k} - \sqrt{n} \leq \frac{k}{\sqrt{n}}. $$ I can Taylor expand roughly, but I would like to make this rigorous.

Answer 1

\begin{align} \sqrt{n+k}-\sqrt{n} = \frac{k}{\sqrt{n+k}+\sqrt{n}} \le \frac{k}{\sqrt{n}} \end{align}

where the last inequality is due to $\sqrt{n+k}$ is positive and $k>0$.