The stability analysis of nonlinear continuous systems often requires manual calculation, which can become time-consuming when dealing with complex systems. Some works use positive invariant sets to discuss stability. These sets can be numerically approximated using Interval analysis but the computational complexity is exponential. In this paper, we propose a computational tractable numerical but guaranteed method based on Interval analysis to verify the robust positive invariance of ellipsoids to automatize the study of n-dimensional nonlinear systems' stability. This method relies on a fast enclosure of a state integration by an Euler method. Interval analysis guarantees the results of the developed algorithms. Several examples show the effectiveness of the proposed approaches on $n$-dimensional non-asymptotical continuous systems subject to bounded perturbation.