# Another sequence convergence proof

1. Sep 27, 2007

### antiemptyv

1. The problem statement, all variables and given/known data

Let $$y_n := \sqrt{n+1} - \sqrt{n}$$ for $$n \in \mathbb{N}$$. Show that $$(y_n)$$ converges.

2. Relevant equations

3. The attempt at a solution

I see that it converges to 0. I just need a nudge in the right direction at getting into $$| \sqrt{n+1} - \sqrt{n} - 0 | = | \sqrt{n+1} - \sqrt{n} |$$ to show it's less than any $$\epsilon > 0$$. Any manipulating I've tried so far makes the terms way too big to work with.

Last edited: Sep 27, 2007
2. Sep 27, 2007

### morphism

$$\left(\sqrt{n+1} - \sqrt{n}\right) \cdot \frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}}$$

3. Sep 27, 2007

### antiemptyv

ohhhh, i see it now.

4. Jan 21, 2009

### ynotidas

What do you do after
1/(sqrt{n+1)+sqrt{n}) ??