Hilbert's basis theorem

Question

In first proof of wikipedia:

Why the $deg(f_i)$ is a non-decreasing sequence of naturals?

Answer 1

$f_n$ and $f_{n+k}$ are both polynomials in $\mathfrak a\setminus\mathfrak b_n$ and, since $f_n$ has minimal (technically, least) degree, $\deg f_n\le \deg f_{n+k}$.