Proving Factorial Greater Than Exponential Using Induction

Prove that $n! > 2^n$ for $n \geq 4$ using mathematical induction.

This problem is an excellent demonstration of mathematical induction, a fundamental proof technique in discrete mathematics. The statement involves proving that for every natural number n greater than or equal to 4, the factorial of n is strictly greater than two raised to the power of n. The use of induction requires us to first verify the base case, which is typically the smallest number for which the statement claims to be true. In this problem, our base case is n equals 4. Factoring in the factorial's inherent exponential growth compared to the linear rate of the exponential function with base 2 is crucial in understanding why the statement holds.

Induction is formally structured with a base case and an inductive step. Once the base case is verified, the inductive step assumes that the statement is true for some arbitrary natural number k and then shows that this assumption implies the statement is true for k plus one. This logical step is what allows us to propose that since the statement holds for the base case and the inductive step follows, it holds for all numbers greater than or equal to our base case.

This problem ties into broader concepts such as the comparison of growth rates of functions, where factorial functions grow significantly faster than exponential functions, a concept often touched upon in algorithm analysis. Understanding the disparity in these growth rates is key when considering algorithm efficiencies, especially in computer science. The proof also touches on basic function growth comparisons by using induction to logically predict a trend beyond computational verification.