Prove $\sqrt{n+1} - \sqrt{n} \to 0$ as $n \to \infty$

Question

As in the title:

Prove $\sqrt{n+1} - \sqrt{n} \to 0$ as $n \to \infty$

It seems so simple, but I can't seem to do it without ending up in circular reasoning.

Answer 1

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

Answer 2

you can rewrite your expression as: $$ \frac{n^{1/2}}{n} \frac{(1+\frac{1}{n})^{1/2}-1}{\frac{1}{n}} $$. As n goes to $\infty$ the expression of the secon fraction is a known limit that is equal to $1/2$. So you get as n goes to $\infty$ $1/2 \frac{1}{\sqrt{n}}$ that goues to zero.