This paper presents a new lemma for the Caputo fractional derivative when 0 < α < 1, which allows finding a simple Lyapunov candidate function for many fractional order systems, and consequently proving their stability using the fractional-order extension of the Lyapunov direct method. The lemma provides a useful inequality that simplifies the analysis of stability for fractional order systems. The paper also demonstrates the usefulness of this lemma through two examples: one for a fractional order linear time-varying system and another for a fractional order nonlinear system. The results show that the lemma can be used to prove the stability of these systems, and the examples confirm the effectiveness of the method. The paper concludes that the proposed lemma is a valuable tool for analyzing the stability of fractional order systems.This paper presents a new lemma for the Caputo fractional derivative when 0 < α < 1, which allows finding a simple Lyapunov candidate function for many fractional order systems, and consequently proving their stability using the fractional-order extension of the Lyapunov direct method. The lemma provides a useful inequality that simplifies the analysis of stability for fractional order systems. The paper also demonstrates the usefulness of this lemma through two examples: one for a fractional order linear time-varying system and another for a fractional order nonlinear system. The results show that the lemma can be used to prove the stability of these systems, and the examples confirm the effectiveness of the method. The paper concludes that the proposed lemma is a valuable tool for analyzing the stability of fractional order systems.