Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction

Does it have to be full induction? Because it seems to be way simpler:

$$(1)\;\;\;\;\;\;2\left(\sqrt{n+1}-\sqrt n\right)=\frac{2}{\sqrt{n+1}+\sqrt n}\leq\frac{2}{\sqrt n+\sqrt n}=\frac{1}{\sqrt n}$$

$$(2)\;\;\;\;\;\;2\left(\sqrt n-\sqrt{n-1}\right)=\frac{2}{\sqrt n+\sqrt{n-1}}\geq\frac{2}{\sqrt n+\sqrt n}=\frac{1}{\sqrt n}$$