Given information:
f and g are real-valued functions defined on same non-negative integers and f(n)≥0,g(n)≥0 for all n≥r and r is any real number greater than zero.
Formula used:
Let f and g be real valued functions defined on the same nonnegative integers, with g(n)≥0 for every integer n≥r, where r is positive real number.
Then,
f is of order g, written f(n) is Θ(g(n)), if and only if, there exist positive real numbers A,B and k≥r such that
Ag(n)≤f(n)≤Bg(n) for every integer n≥k.
Proof:
If f(n) is Θg(n), according to the definition of the Θ notation, there exist A,B and k positive real numbers with