Prove that big O notation forms an equivalence relation on F(R+,R+), the set of functions from R+ to itself. We will say that if limx f(x)/g(x) = 0...

1. Prove that big O notation forms an equivalence relation on F(R+,R+), the set of functions from R+ to itself.2. We will say that if limx→∞ f(x)/g(x) = 0 then O(f) < O(g). From this definition, show that if O(f) = O(f′),O(g) = O(g′) and O(f) < O(g) then O(f′) < O(g′).

1. Prove that big O notation forms an equivalence relation on F(R+,R+), the set offunctions from R+ to itself.Proof. Reflexive: Suppose x ∈ R. Then x − x = 0, which is an integer. Thus, xRx....