Prove that between any two different real numbers there is a rational number and an irrational number.
My attempt at a proof: (Throughout this proof it is assumed without any loss of generality that $x < y$)
1.) $x$ is irrational and $y$ is irrational.
In this case the arithmetic mean $\frac{x+y}{2}$ is always an irrational number between $x$ and $y$. Also, if $y$ is irrational then it has a non-terminating decimal expansion and so we can always find a positive integer $n$ such that $x < y - 10^{-n}y < y$. The middle of the three numbers here will have a terminating decimal expansion, and hence it is rational. It also clearly lies between $x$ and $y$. These examples of rational and irrational numbers should also work in the cases:
2.) $x$ is irrational and $y$ is rational.
3.) $x$ is rational and $y$ is irrational.
There is one last case to consider.
4.) $x$ is rational and $y$ is rational.
In this case the arithmetic mean of $x$ and $y$ is rational. An irrational number for example could be the ratio $\frac{ex + \pi y}{e + \pi}$
I'm not sure if this proof is a good method or if it is even valid, so some feedback would be appreciated.