True or False? For each part: If true, prove it carefully. You can apply or cite any of our relevant definitions, theorems, examples, or exercises in a proof. If false, provide a counterexample and explain it.

(a) If {A_n : n N} is any collection of subsets of R, with each set A_n containing a finite number of values, then the union n=1 to infinityA_nis closed and bounded.

(b) If (a_n) is any Cauchy sequence of real numbers and (b_n) is any increasing sequence of negative numbers, then the sequence (a_n - 2b_n) converges.