Example

Prove that the product of two consecutive odd numbers is always odd

Let $a$ and $b$ be the two consecutive odd numbers. 1.1. Without loss of generality, assume that $a < b$ , hence $b = a + 2$ 1.2. Now, $a = 2 k + 1$ for some integer $k$ (by definition of odd numbers) 1.3. Similarly, $b = a + 2 = 2 k + 3$ 1.4. Therefore, $ab = (2 k + 1) (2 k + 3) = 4 k^{2} + 8 k + 3 = 2 (2 k^{2} + 4 k + 1) + 1$ (by basic algebra) 1.5. let $m = (2 k^{2} + 4 k + 1)$ which is an integer (by closure of integers under addition and multiplication) 1.6. Then, $ab = 2 m + 1$ , which is odd (by definition of odd numbers)
Therefore, the product of two consecutive odd numbers is always odd

WLOG

“Without loss of generality” may be abbreviated to WLOG

This is used before an assumption in a proof which narrows the premise to some special case
It implies that the proof of this special case can be easily applied to all other cases