Is the following statement about rationals true or false?

Question

I would like to know whether the following statement is true or false?

Suppose $A>B$, then there is a rational number between $A$ and $B$.

I need a proof if the statement is true. Please somebody explain to me how to do this problem. Thank you.

Answer 1

$\mathbb{Q}$ is dense in $\mathbb{R}$, so between $A$ and $B$ there is a rational number.

Choose $n\in \mathbb{N}$ with $$\frac{1}{n}<A-B.$$ and $k=\min\{m\in\mathbb{N}\mid \frac{m}{n}>B\}$. Then $B<\frac{k}{n}<A$.