In the following text we prove that if X is the Alexandroff compactification of a discrete space with at least two elements and f : X → X is a homeomorphism, then in the dynamical system (X, f) the following statements are equivalent: • the dynamical system (X, f) is proximal, 26 Fatemah Ayatollah Zadeh Shirazi et al. • the dynamical system (X, f) is strongly proximal, • the induced hyperspace dynamical system (P 0 (X), f |P<ω 0 (X)) is proximal (where P 0 (X) is the collection of all finite subsets of X equipped with Vietoris topology). Moreover, (X, f) is distal if and only if (P 0 (X), f |P<ω 0 (X)) is distal. The final Section is dedicated to more details on induced hyperspace dynamical systems. Mathematics Subject Classification: 37B05, 54B20, 54H20