Can someone tell me why we can assume that the chains of subspaces become constant for large $r$ ?
It is page 122 of the 2nd edition of the book
Here is also a pdf file:
http://home.ustc.edu.cn/~liweiyu/documents/Algebra,%20Second%20Edition,%20Michael%20Artin.pdf
Because $V$ is finite dimensional, $K_i$ and $U_i$ must also be finite dimensional. If, for instance, the sequence of $K_i$ did not become eventually constant, then there is some sub-sequence $K_{i'}$ that is strictly increasing. This means that the dimensions of $K_{i'}$ are also strictly increasing. However, each of these must also be bounded above by the dimension of $V$. So, we have a strictly increasing set of integers that has an upper bound, a contradiction.
The proof for the $U_i$ is similar, replacing "increasing", "upper", and "above" with their antonyms as appropriate.