This work is dedicated to the stability analysis of time-delay systems with a single constant delay. To answer this problem, Lyapunov-Krasovskii methods have been widely used in the literature and numerous sufficient conditions of stability are proposed and expressed as linear matrix inequalities (LMIs). These conditions being only sufficient, these contributions are often criticized because of the lack of information regarding the reduction of the conservatism. Recently, scalable methods have been investigated using Bessel-Legendre inequality or orthogonal polynomial-based inequalities. The interest of these methods relies on their hierarchical structure. However, the convergence is still an open question that will be answered for the first time in this paper. The objective of this paper is thus to prove that, the stability of a time-delay system implies the feasibility of the scalable LMIs provided in Seuret et al., at a sufficiently large order of the Bessel-Legendre inequality and an estimation of this order is provided analytically.