This paper considers the problem of stability analysis of discrete-time dynamics that are positively homogeneous of degree one. An example of a homogeneous and even continuous dynamics that is globally exponentially stable and that does not admit any λ-contractive proper C-set is presented. This motivates us to propose a natural generalization of this concept, namely, (k, λ)-contractive proper C-sets. It is proven that this simple generalization yields a non-conservative Lyapunov-type tool for stability analysis of homogeneous dynamics, namely, sublinear finite-time Lyapunov functions. Moreover, scalable and non-conservative stability tests are established for relevant classes of homogeneous dynamics.