I don't get why the interval is divided into three pieces. Wouldn't the same reasoning work if the interval were divided into just two pieces and you always select the sub-interval not containing the Nth algebraic number?
In fact I've recently thought that one could divide it into 5. Then you always discard the outer divisions, and choose one of the inner divisions that doesn't contain the algebraic you're avoiding. In this way the leftmost endpoints of the chosen intervals form a strictly increasing sequence that's bounded above, and the rightmost endpoints of the chosen intervals form a strictly decreasing bounded sequence. That helps you to see that you really want these things to have limits.
Does that help? If you ask more questions I can improve the article.
Thanks.