pattern in a certain approximation of square roots

Question

I was playing around with approximating an irrational number $x>1$ as the product of an integer and terms of the form $\frac {n+1}{n}, n\in \mathbb{N}$. In particular I investigated the sequence which forms when you choose each $n$ using a greedy algorithm for the best approximation you can get without going over. To do this, you wind up taking $x$, and dividing by your approximation so far, call this answer $y$. Then you take $\frac {1}{y-1}$ and round up to get the next $n$ in the sequence. running this with square root 3, I noticed that as you go, $\frac {1}{y-1}$ seems to tend in the limit towards being an integer plus $0.5$. The pattern also holds true as far as I've checked for $\sqrt{2},and \sqrt 5$. Can someone explain this pattern?