Prove that there is no largest irrational number

Question

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.

Answer 1

Assume there is one. Add 1. QED.

Answer 2

Hint: $n+\sqrt 2$ is irrational for all $n \in \mathbb N$.

Answer 3

A direct proof is probably simplest.

Given any irrational number $\alpha,$ we have by the previous result that $\alpha+1$ is also irrational. Since $\alpha<\alpha+1,$ then $\alpha$ cannot be the greatest irrational number. Since $\alpha$ was an arbitrary irrational number, then there is no such thing as "the greatest irrational number" at all.