Let A ⊆ Z a subset of the integers such that satisfies the following properties.?
1) If n, l ∈ A then n + l ∈ A.
2) If n ∈ A and k ∈ Z then kn ∈ A.
a) Let l ∈ N fixed and consider the set B {kl = k ∈ Z}, show that B satisfies properties 1) and 2).
b) Let A ⊆ Z nonempty satisfying such properties 1) and 2). Prove that 0 ∈ A if A has more than an element then has positive elements.
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a) Take any elements kl and hl from B, and any integer z ∈ Z.
1) kl + hl = (k+h)l lies in B, since k+h ∈ Z.
2) z(kl) = (zk)l lies in B, since zk ∈ Z.
b) Let n be any element of A. Since 0 ∈ Z, property (2) assures us that 0 = 0n ∈ A.
If A contains more than one element, one such element n will be non-zero. Either n itself is positive, or, with (-1) ∈ Z, the number -n = (-1)n ∈ A must be. So A definitely contains at least one positive element.