<s>[INST] Proving $\sqrt{n}+\frac{1}{\sqrt{n+1}} \geq \sqrt{n+1}$ I would like to know how to prove the following assertion :

For every $n>0$: $$\sqrt{n}+\frac{1}{\sqrt{n+1}} \geq \sqrt{n+1}$$

 [/INST] Or you could obfuscate the problem. Let $t=\sqrt{n+1}$. Then we have $t>1$ and $\sqrt{n}=\sqrt{t^2-1}$, so
$$
\begin{align*}
 &\sqrt{t^2-1}+\frac{1}{t} \ge t\\
\text{iff}\qquad &\sqrt{t^2-1}\ge t-\frac{1}{t}\\
\text{iff}\qquad &t^2-1 \ge t^2-2+\frac{1}{t^2}\\
\text{iff}\qquad &1 \ge \frac{1}{t^2}\\
\text{iff}\qquad &t^2\ \ge 1
\end{align*}
$$
which is true, since $t>1$.
</s>