Prove that the difference between two rational numbers is rational

This is a terribly simple question I’m sure, but I can’t find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt:

Let $a$ and $b$ be rational numbers. Therefore, \begin{align}
a=\frac{\lambda}{\beta},\:b=\frac{\xi}{\zeta},\:\ni\:\lambda,\beta,\xi,\zeta\in\mathbb{Z},\tag{1} \end{align} which gives us \begin{align}
a-b=\frac{\lambda}{\beta}-\frac{\xi}{\zeta}=\frac{\lambda\zeta-\xi\beta}{\beta\xi}.\tag{2}
\end{align}

So I have shown that $a$ and $b$ are rational numbers which, by definition, can be represented by the quotient of two integers. But now how do I tackle the problem of the difference? By definition the difference between two integers is an integer. Does that require that this difference is thus rational?

Thank you for your time,

Solutions Collecting From Web of "Prove that the difference between two rational numbers is rational"

$$\begin{align*}
\lambda &\color{red}\leftarrow \color{red}{\textrm{integer}} \\
\beta &\color{red}\leftarrow \color{red}{\textrm{integer}} \\
\xi &\color{red}\leftarrow \color{red}{\textrm{integer}} \\
\zeta &\color{red}\leftarrow \color{red}{\textrm{integer}} \\
\alpha\zeta &\color{red}\leftarrow \color{red}{\textrm{integer}} \\
\xi\beta &\color{red}\leftarrow \color{red}{\textrm{integer}} \\
\alpha\zeta – \xi\beta &\color{red}\leftarrow \color{red}{\textrm{integer}} \\
\beta\zeta &\color{red}\leftarrow \color{red}{\textrm{integer}} \\
\frac{\alpha\zeta-\beta\xi}{\beta\zeta} &\color{red}\leftarrow \color{red}{\frac{\textrm{integer}}{\textrm{integer}}}
\end{align*}$$