Monotonic sequence and limit 
For $x_n = \frac{1}{2+x_{n-1}}$ where $x_1 =1/2$, show that the
  sequence is monotonic and find its limit.

What I first did was finding $x_{n+1}$, which equals $\frac{1}{2+x_n}$; then $x_{n+2}=\frac{1}{2+\frac{1}{2+x_n}}=\frac{x_n+2}{2x_n+5}$ thus it does eventually get smaller hence $x_n>x_{n+1}$. How can I finish this? 

$
x_\infty=\frac{1}{2+x_\infty} \Rightarrow x_\infty^2+2x_\infty-1=0 \Rightarrow x_\infty= \sqrt2-1$ as summation of positive terms can not be negative
