Cauchy sequence $x_n=sqrt{a+x_{n-1}}$

I have to show that this sequence
$$

x_n=\sqrt{a+x_{n-1}} \hbox{ with } x_1=\sqrt{a}
$$
is a Cauchy sequence for every $a>0$. I have done the following calculations:
$$
\left| x_{n+2}-x_{n+1} \right|=\left| \sqrt{a+x_{n+1}}-x_{n+1}\right|=\left|\frac{a+x_{n+1}-x_{n+1}^2}{\sqrt{a+x_{n+1}}+x_{n+1}} \right|= \left|\frac{x_{n+1}-x_n}{\sqrt{a+x_{n+1}}+x_{n+1}}\right|
$$
I don't come up with a boundary for the denominator so that
$$
\left|\frac{x_{n+1}-x_n}{\sqrt{a+x_{n+1}}+x_{n+1}}\right|$$

Any hint? I know I can use induction to easily prove that the sequence is convergent, but I'd like to prove it's Cauchy without using convergence. Thank you in advance.