Given information:
The functions f1, f2 and g are defined on same set of non-negative integers and for every n≥r, f1(n)≥0, f2(n)≥0 and g(n)≥0 where r is a positive real number.
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 at least g, written f(n) is Θ(g(n)), if and only if, there exist positive real numbers A,B and a≥r such that
Ag(n)≤f(n)≤Bg(n) for every integer n≥a.
Proof:
Using the definition of Θ− notation,
there exists four real numbers A1≥0,A2≥0,B1≥0 and B2≥0 such that A1g(n)≤f1(n)≤B1g(n) and A2g(n)≤f2(n)≤B2g(n) for every n≥r