I'm not able to find a limit of a sequence.

It's related to the Babylonian's method to approximate a square root. In our case, we have $\displaystyle a>0$, and we want to approximate $\displaystyle \sqrt a$. Thanks to the Newton's method I've found the Babylonian's method. We have $\displaystyle x_{n+1}=\frac{x_n+\frac {a}{x_n}} {2}$. We can start the method for any $\displaystyle x_0>0$. I also showed that the sequence $\displaystyle x_n$ is decreasing and I must assume (since I wasn't able to demonstrate it) that it is bounded below by $\displaystyle \sqrt a$. Now I must prove that the sequence $\displaystyle {x_n}$ converges to $\displaystyle \sqrt a$ when $\displaystyle n$ tends to $\displaystyle +\infty$ and this is precisely what I'm asking you to help me.