The one-to-one relation between the winding number and the number of robust zero-energy edge states, known as bulk-boundary correspondence, is a celebrated feature of 1d systems with chiral symmetry. Although this property can be explained by the K-theory, the underlying mechanism remains elusive. Here, we demonstrate that, even without resorting to advanced mathematical techniques, one can prove this correspondence and clearly illustrate the mechanism using only Cauchy's integral and elementary algebra. Additionally, our approach to proving bulk-boundary correspondence sheds light on a kind of system that doesn't respect chiral symmetry but has robust left or right zero-energy edge states, where one can still assign the winding number to characterize these zero-energy edge states.
