Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

Beginning Activity Beginning Activity 1: Exploring a Proposition about Factorials

A4US

Using this definition, we see that

In general, we write $n!=1\cdot 2\cdot 3\cdots \left(n-1\right)\cdot n$ or $n!=n\cdot \left(n-1\right)\cdots 2\cdot 1\text{.}$ Notice that for any natural number $n\text{,}$ $n!=n\cdot (n-1)!\text{.}$

Now let $P(n)$ be the open sentence, “$n!>2^n\text{.}$”

Let $k$ be a natural number with $k\ge 4\text{.}$ Suppose that we want to prove that if $P(k)$ is true, then $P(k+1)$ is true. (This could be the inductive step in an induction proof.) To do this, we would be assuming that $k!>2^k$ and would need to prove that $(k+1)!>2^{k+1}\text{.}$ Notice that if we multiply both sides of the inequality $k!>2^k$ by $(k+1)\text{,}$ we obtain