The trapezoidal formula is noted for having the smallest truncation error among linear multistep methods with a specific stability property. The paper derives error bounds for this method under general conditions, ensuring that the error remains bounded as \( t \to \infty \), even when the product of the Lipschitz constant and the step-size is large. The stability requirement is that the integral curve is uniformly asymptotically stable in the sense of Liapunov. The introduction provides a detailed definition of the general linear \( k \)-step method for approximating solutions to first-order ordinary differential equations. The method is defined by a recurrence relation involving coefficients \( \alpha_i \) and \( \beta_i \). The theory of such methods is well-established, and the paper discusses the conditions for explicit and implicit methods. The order of the method is determined by the largest integer \( p \) such that the operator \( L \) annihilates the polynomial \( \varrho(t) \). Consistency is ensured if \( p \geq 1 \), and the error constant \( c^* \) is used to compare the accuracy of methods with the same order. The paper also explores the relationship between the polynomials \( \varrho(\zeta) \) and \( \sigma(\zeta) \) and the error constant, and discusses the limitations on the maximum order \( p \) due to stability requirements.The trapezoidal formula is noted for having the smallest truncation error among linear multistep methods with a specific stability property. The paper derives error bounds for this method under general conditions, ensuring that the error remains bounded as \( t \to \infty \), even when the product of the Lipschitz constant and the step-size is large. The stability requirement is that the integral curve is uniformly asymptotically stable in the sense of Liapunov. The introduction provides a detailed definition of the general linear \( k \)-step method for approximating solutions to first-order ordinary differential equations. The method is defined by a recurrence relation involving coefficients \( \alpha_i \) and \( \beta_i \). The theory of such methods is well-established, and the paper discusses the conditions for explicit and implicit methods. The order of the method is determined by the largest integer \( p \) such that the operator \( L \) annihilates the polynomial \( \varrho(t) \). Consistency is ensured if \( p \geq 1 \), and the error constant \( c^* \) is used to compare the accuracy of methods with the same order. The paper also explores the relationship between the polynomials \( \varrho(\zeta) \) and \( \sigma(\zeta) \) and the error constant, and discusses the limitations on the maximum order \( p \) due to stability requirements.