Show that every positive integer is either even or odd.

To prove that every positive integer is either even or odd, let's start by defining what we mean by "even" and "odd."

Even number: A number is even if it is divisible by 2. In other words, an integer n is even if there exists an integer k such that: n = 2k

Odd number: A number is odd if it is not divisible by 2. In other words, an integer n is odd if there exists an integer k such that: n = 2k + 1

Proof:
Let n be any positive integer. We will prove that n must either be even or odd.

If n is divisible by 2, then by the definition of even numbers, n is even.

If n is not divisible by 2, then there must exist an integer k such that n = 2k + 1. This is the definition of an odd number.

Thus, for every positive integer n, either:

n is even (i.e., n = 2k for some integer k), or
n is odd (i.e., n = 2k + 1 for some integer k).
This shows that every positive integer is either even or odd.

Therefore, we have proven the statement.