We give a constructive proof of the finite version of Gowers' \(\mathrm{FIN}_k\) Theorem and analyse the corresponding upper bounds. The \(\mathrm{FIN}_k\) Theorem is closely related to the oscillation stability of \(c_0\). The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was proved well before by V. Milman. We compare the finite \(\mathrm{FIN}_k\) Theorem with the Finite Stabilization Principle found by Milman in the case of spaces of the form \(l_\infty^n\), for \(n \in \mathbb{N}\), and establish a much slower growing upper bound for the finite stabilization principle in this particular case.