  Photon sphere has attracted significant attention since the discovery of
black hole shadow images by Event Horizon Telescope. Recently, a number of
studies have highlighted that the number of photon spheres and their
distributions near black holes are strongly constrained by black hole
properties. Specifically, for black holes with event horizons and proper
asymptotic behaviors, the number of stable and unstable photon spheres
satisfies the relation $n_{\text{stable}} - n_{\text{unstable}} = -1$. In this
study, we provide a novel proof on this relation using a completely new
geometric approach, in which the stable and unstable photon spheres are
determined by Gaussian curvature and geodesic curvature in the optical geometry
of black hole spacetimes. Firstly, we demonstrated the existence of photon
spheres near black holes with proper asymptotic behaviors (asymptotically flat
black holes, asymptotically de-Sitter and anti-de-Sitter black holes).
Subsequently, we proved that the stable and unstable photon spheres near black
holes must be one-to-one alternatively separated from each other, such that
each unstable photon sphere is sandwiched between two stable photon spheres
(and each stable photon sphere is sandwiched between two unstable photon
spheres). Our analysis in this study is limited to the spherically symmetric
black hole spacetime (with the spacetime metric
$ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$).
