Consider two functions with natural arguments and positive real values: f; g : N â R+. Consider the following propositions, where c ranges over positive reals, and n, n0 range over the naturals: P : âc : ân0 : ân : n ⥠n0 ! f(n) ⤠cg(n) (this is the de nition of \f(n) is O(g(n))")
Q : ân0 : âc : ân : n ⥠n0 ! f(n) ⤠cg(n)
R : âc : ân0 : ân : n ⥠n0 ! f(n) â¤cg(n)

Required:
a. Are any two of these propositions equivalent, for arbitrary choices of f and g?
b. Is any of the propositions always true, whatever f and g?
c. Can you nd two functions f and g for which R is true ?