Kronecker’s theorem

Lemma 3. Theorem 1 is true if and only if Theorem 2 is true. Proof. Assume that Theorem 2 is true and let θ′ 1, . . . , θ ′ k, 1 be real numbers that are linearly independent over Z, let α1, . . . , αk be real numbers, let N > 0 and let 0 < < 1. Let θm = θ ′ m − qm with 0 < θm ≤ 1. Because θ′ 1, . . . , θ′ k, 1 are linearly independent over Z, so are θ1, . . . , θk, 1. Using Theorem 2 with k + 1 instead of k, N + 1 instead of T , 12 instead of , applied with θ1, . . . , θk, 1, α1, . . . , αk, 0, 1K. Chandrasekharan, Introduction to Analytic Number Theory, pp. 92–93, Chapter VIII, §5.