Between Any Two Real Numbers Is Another Real Number

Let $a$ and $b$ be distinct real numbers with $a<b$. We need to produce a number $c$ that lies between $a$ and $b$. Let $c=\frac{a+b}{2}$ (which is the average of $a$ and $b$). We need to show that $a<c$ and that $c<b$. Notice that $$a=\frac{a+a}{2}<\frac{a+b}{2} =c,$$ where the middle inequality holds because $a<b$. Similarly, $$c=\frac{a+b}{2}<\frac{b+b}{2} =b.$$ This proves there exists a real number $c$ between $a$ and $b$.