Square root inequality

Note that $\left(\sqrt{n+1}-\sqrt{n}\right)\cdot \frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}} = \frac{(n+1)-n}{\sqrt{n+1}+\sqrt{n}}=\frac{1}{\sqrt{n+1}+\sqrt{n}}<\frac{1}{\sqrt{n}}$ which is always true for all real $n\geqslant1$ .

$\sqrt{n+1}-\sqrt{n}<\frac{1}{\sqrt{n}}\\ \sqrt{n^2+n}-n<1=(n+1)-n=\sqrt{n^2+2n+1}-n\\ \sqrt{n^2+n}<\sqrt{n^2+2n+1} \\ n^2+n<n^2+2n+1\\ 1<n$

And as $n>101>1$ the inequality holds and the claim is false.