In this paper we define the notion of the lower sensitivity for the pseudo-metrics. We evaluate the lower sensitivities of the Schwarzschild metric when the redial coordinate is near the event horizon. In the black-hole case we find finite lower sensitivity in the direction of time, and infinite lower sensitivities in the other directions. The finiteness of the lower sensitivity of the black hole implies the finiteness of its entropy, which confirms "the recent physical results on black-hole gravitational entropy" mathematically.