[Math] Show that $\lim_{n\to \infty} (\sqrt{n+1} – \sqrt n) = 0$ using the definition of a limit

Question

I found that the $\lim_{x\to \infty}\left( \sqrt{n+1}-\sqrt{n}\right )=0$. But I have to use the definition of a limit to solve too.

So far I have given $\epsilon>0, |\sqrt{n+1}-\sqrt{n}|< \epsilon.$ And now I don't know what to do.

Answer 1

Use that $$\sqrt{n+1}-\sqrt{n}=(\sqrt{n+1}-\sqrt{n})\dfrac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}=\dfrac{1}{\sqrt{n+1}+\sqrt{n}}\leqslant \dfrac{1}{2\sqrt{n}}$$