Does anyone know an "elementary" proof of the fact that a Noetherian local ring has finite Krull dimension?
The one I know is from Atiyah&Macdonald's book Introduction to Commutative Algebra, where they use Hilbert functions (which is not an elementary proof). On the other hand, I am studying the local rings section of Weibel's book Introduction to Homological Algebra, where it says that if $R$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $k$ then the Krull dimension of $R$ is bounded by $\dim_k\mathfrak{m}/\mathfrak{m}^2$, and there is not any reference about that. This made me think that this could be easy to prove but I haven't succeeded.