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## Homework Statement

Show that ##\displaystyle \lim_{n\to \infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n}) = \frac{1}{2}##

## Homework Equations

## The Attempt at a Solution

We see that ##\displaystyle \sqrt{n}(\sqrt{n+1}-\sqrt{n}) - \frac{1}{2} = \frac{\sqrt{n}}{\sqrt{n+1}+\sqrt{n}} - \frac{1}{2} < \frac{\sqrt{n}}{2\sqrt{n}} - \frac{1}{2} = \frac{1}{2} - \frac{1}{2} = 0##. But I don't think this can be right... What am I doing wrong?