Another proof of a Gowers theorem

W. T. Gowers proved that every Lipschitz function from the unit sphere of the Banach space $c_0$ to $\mathbb{R}$ is oscilation stable. His proof uses a result about finite partitions of the set $FIN_k$ of finitely supported functions $p$ from $\mathbb{N}$ to $\{0,1,...,k\}$ with $k$ in $Im(p)$. Every known proof of this fact uses methods of topological dynamics on the space $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$. We give a purely combinatorial proof of this result avoiding the use of ultrafilters.