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.