Math Help - Real Analysis Sequence Question

1. Real Analysis Sequence Question

Im not sure how to post the picture here so I have as a attachment

2. This is just a good ole induction problem for the decreasing/bounded part.
The because bounded monotone sequences converge you have to solve $L = \frac{1}{2}\left( {L + \frac{{a^2 }}{L}} \right)$ for $L$.