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 , and we want to approximate . Thanks to the Newton's method I've found the Babylonian's method. We have . We can start the method for any . I also showed that the sequence is decreasing and I must assume (since I wasn't able to demonstrate it) that it is bounded below by . Now I must prove that the sequence converges to when tends to and this is precisely what I'm asking you to help me.