I have to prove that there is no largest irrational number from the result of the a previous proof: Prove that if $x$ is rational and $y$ is irrational then, $x+y$ is irrational. I was able to prove this so I'll put it here.
Proof: Assume to the contrary that since $x$ is rational and $y$ is irrational then $x+y$ is a rational number $z$. Thus $x+y=z$, where $x=\frac{a}{b}$ and $z=\frac{c}{d}$ for some integers $a,b,c,d \in \mathbb{Z}$ and $b,d \neq 0$. This implies that $y=\frac{c}{d}-\frac{a}{b}=\frac{bc-ad}{bd}$.
Since $bc-ad$ and $bd$ are integers and $bd \neq 0$, it follows that $y$ is rational, which is a contradiction.
Not sure how to go about proving that there is no largest irrational number.
Best Answer
Assume there is one. Add 1. QED.