Prove each of the following properties of big oh:

a. f(n) = O(f(n)).

b. If f(n) = O(g(n)) and g(n) = O(h(n)), then f(n) = O(h(n)).

c. If f(n) = O(g(n)), then af(n) = O(g(n)) for any real number a.

d. If f(n) = O(g(n)), then af(n) = O(ag(n)) for any number a.

e. If f(n) = O(g(n)), then f(n/b) = O(g(n/b)) for b > 0.

f. If f₁(n) = O(g₁(n)) and f₂(n) = O(g₂(n)), then f₁(n)f₂(n) = O(g₁(n)g₂(n)).

g. If f₁ and f₂ have nonnegative values and f₁(n) = O(g₁(n)) and f₂(n) = O(g₂(n)), then f₁(n) + f₂(n) = O(g₁(n) + g₂(n)).

h. If f and g have positive values and f(n) = O(g(n)), then 1/g(n) = O(1/f(n)).