Factorials Greater Than Powers of Two

Prove that for any integer $n > 4$, $n! > 2^n$.

This problem is an exercise in understanding factorials and how rapidly they grow compared to powers of two. The task is to prove that for any integer greater than 4, the factorial of that integer is greater than two raised to the power of that integer. Factorials, denoted by the exclamation mark following an integer, such as $n!$, are the product of all positive integers less than or equal to the integer $n$. This function grows extremely quickly because you are successively multiplying numbers together. In contrast, an exponential function like $2^n$ grows by repeatedly multiplying by a constant base, which is generally slower compared to factorial growth except for small values of $n$.

To prove this inequality, the technique of mathematical induction is highly effective. Induction is a proof strategy that works particularly well with statements that are labeled by natural numbers. The basic idea is to first show that the statement holds for an initial case, usually $n = 5$ in problems like this since we start at $n > 4$. Then, you assume it holds for an arbitrary integer $k$, and subsequently prove that it must then hold for the next integer, $k+1$. Successfully doing this establishes the truth of the statement for all integers greater than your starting case. This type of task not only strengthens one's skills in manipulating inequalities and understanding factorial functions but also provides insight into proof techniques crucial in discrete mathematics, particularly in sequences and induction.