Decide which of the following statements are true and which are false. Prove the true ones and provide a counterexample for the false ones.
a) If {xn} is Cauchy and {yn} is bounded, then {xnyn} is Cauchy.
b) If {xn} and {yn} are Cauchy and yn ≠ 0 for all n ∊ N, then {xn/yn} is Cauchy.
c) If {xn} and {yn} are Cauchy and xn + yn > 0 for all n ∊ N, then {1/(xn + yn)} cannot converge to zero.
d) If [xn] is a sequence of real numbers that satisfies x2k - x2k-1 → 0 as k → ∞ and if xn = 0 for all n ≠ 2k, k ∊ N, then {xn} is Cauchy.