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So I started as such:

WLOG let $a,b$ be irrational numbers such that $a<b$. By theorem (not sure if there is a name for it), we know that there exist a rational number $x$ such that $a<x<b$.

If I can show there is another irrational between $a$ and $b$, I could then use the fact that between every two rational numbers there is a rational number and repeated applications would show that there are infinitely many.