Suppose f(n) = O(s(n)), and g(n) = O(r(n)). All four functions are positive-valued and monotonically increasing. Prove (using the formal definitions of asymptotic notations) or disprove (by counterexample) each of the following claims:

(a) f(n) g(n) = O(s(n) r(n))

(b) if s(n) = O(g(n)), then f(n) = O(r(n))

(c) if r(n) = O(s(n)), then g(n) = O(f(n))

(d) if s(n) + g(n) = O(f(n)), then f(n) = (s(n))