Hi, I know how to show the following sequence goes to its limit, but i need to do it using the definition, of which i am having trouble, i've started how i should though i think.

$\displaystyle x_{n} = \sqrt{n+1} - \sqrt{n}$

prove $\displaystyle x_{n} \rightarrow 0$

Let $\displaystyle \epsilon > 0$ be given

Let $\displaystyle N=N(\epsilon )$ be an integer greater than ...(answer goes here)...

Then for all $\displaystyle n>N$ we have

$\displaystyle |x_{n} - l| = |\sqrt{n+1}-\sqrt{n} - 0|$

$\displaystyle |x_{n} - l| = |\sqrt{n+1} - \sqrt{n}|$

$\displaystyle \sqrt{n+1} - \sqrt{n} = \dfrac{(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{\sqrt{n+1} + \sqrt{n}}

$

$\displaystyle |x_{n} - l| = \dfrac{1}{\sqrt{n+1} + \sqrt{n}}$

and that is where i get stuck.

I'm not sure if this definition method of proof is well known so if you would like an example of a completed question just ask. Thanks.