  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. Furthermore, our approach to
proving bulk-boundary correspondence also provides clear insights into a kind
of system that doesn't respect chiral symmetry but can have robust left or
right zero-energy edge states. In such systems, one can still assign the
winding number to characterize these zero-energy edge states.
