This study analyzes a Caputo-corrected Grünwald–Letnikov discretization for fractional initial-value problems of order 0 < α < 1. Stability with respect to perturbations in the initial value is established for the linear test equation with λ > 0, showing that such perturbations are not amplified by the time-stepping process. Under stated regularity and consistency assumptions, the numerical solution converges to the exact solution as h → 0. A numerical example with α = 0.8 confirms decreasing absolute error under mesh refinement, from 1.5978 × 10⁻² at h = 0.2 to 6.04 × 10⁻⁴ at h = 0.005.