The first two items of this question are review.
(a) Prove that the composition of one-to-one maps is one-to-one.
(b) Prove that if a linear map is not one-to-one then at least one nonzero vector from the domain maps to the zero vector in the codomain.
(c) Verify the statement, excerpted here, that precedes Theorem 1.8. . . . if a minimal polynomial m(x) for a transformation t factors as m(x) = (x - λ1)q1..... (x - λz)qz then m(t) = (t - λ1)q1 ○.....○ (t - λz)qz is the zero map. Since m(t) sends every vector to zero, at least one of the maps t - λi sends some nonzero vectors to zero. . . . That is, . . . at least some of the λi are eigenvalues.