Prove $\sqrt{n+1}-\sqrt{n}\lt \frac{1}{2\sqrt{n}}$

discrete-mathematics induction

$$\sqrt{n+1}-\sqrt{n} = \frac{\left(\sqrt{n+1}-\sqrt{n}\right)\left(\sqrt{n+1}+\sqrt{n}\right)}{\sqrt{n+1}+\sqrt{n}}=\frac{1}{\sqrt{n}+\sqrt{n+1}}<\frac{1}{\sqrt{n}+\sqrt{n}}.$$


Hint: $$\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}$$


According to the MVT there is $x_0$ between $x$ and $x+1$ such that $$\sqrt{x+1}-\sqrt{x}=\frac{1}{2\sqrt{x_0}}.$$

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