  We generalize the Su-Schrieffer-Heeger (SSH) model with the inclusion of
arbitrary long-range hopping amplitudes, providing a simple framework to
investigate arbitrary adiabatic deformations that preserve the chiral symmetry
upon the bulk energy bands with any arbitrary winding numbers. Using only
elementary techniques of solving linear difference equations and applying
Cauchy's integral formula, we obtain a mathematically rigorous and physically
transparent proof of the bulk-boundary correspondence for the generalized SSH
model. The multiplicity of robust zero-energy edge modes is shown to be
identical to the winding number. On the other hand, nonzero-energy edge modes,
if any, are shown to be unstable under adiabatic deformations and not related
to the topological invariant. Furthermore, under deformations of small spatial
disorder, the zero-energy edge modes remain robust.
