Decide if sequence converges/diverges

The given sequence is: $\displaystyle a_n = n\left(\sqrt{1 + \frac{1}{n}} - 1\right) $.

I rearranged terms to get $\displaystyle a_n = \sqrt{n^2 + n} - n $. I couldn't tell by inspection if this is convergent or divergent, so I started plugging in large values of n and noticed that it seems to be that $\displaystyle a_n \to 0.5$. So now I'm trying to prove it converges by showing that $\displaystyle a_n$ has an upper bound and is non-decreasing. However I'm having trouble showing it has an upper bound, therefore leading me to think maybe that $\displaystyle a_n$ is divergent? I'm not sure at this point, any suggestions would be appreciated.

Thanks in advance.