(a) If Z₁ and Z_2 are null sets, show that Z₁ U Z_2 is a null set.

(b) More generally, if Z_n is a null set for each n∈ N, show that U_{n=1}^∞ Z_n is a null set. [Hint: Given ε > 0 and n∈ N, let {J_{k}^n: k ∈ N} be a countable collection of open intervals whose union contains Z_n and the sum of whose lengths is ≤ ε/2^n. Now consider the countable collection {J_{k}^{n}: n, k ∈ N}.]

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