We prove that the finitistic dimension is bounded for modules of bounded \(k\)-dimension, for all finite dimensional algebras \(\Lambda\) over a field \(k\). Our argument avoids the use of model theory made by Jensen and Lenzing, when they first published this fact. We also show that, given \(d\) and \(m\): there is a number \(f(d,m)\) such that, for any \(d\)-dimensional \(k\)-algebra \(\Lambda\) and any \(\Lambda\)-module \(M\) with \(\dim_k M\leq m\), if \(\text{Ext}^i_\Lambda(M,\Lambda)\neq 0\) for all \(0\leq i\leq f(d,m)\) then \(\text{Ext}^i_\Lambda(M,\Lambda)\neq 0\) for all \(i\). This reveals some ultimate regularity of the injective resolution of the regular module in the general case.