Infinite number of irrational numbers between 0 and 1? Basically, I am asking if this is true.
$$|\{z\mid 0<z<1\land z\in\mathbb{R-Q}\}|=\infty$$
But I don't know how to prove it.
 A: $1/\sqrt{n}$ is irrational for every prime number $n$.  I'm leaving uncountably many out but this proves that there are infinitely many.
A: It's true, and there are many many ways to prove it.
Taking any rational number $q$ such that $0<q<\pi$, the number $\frac{q}{\pi}$ is an irrational number between $0$ and $1$, and since there are infinitely many rationals between $0$ and $\pi$, there must be infinitely many irrationals between $0$ and $1$.

Or, you could say that for every $n$, there exists an irrational number between $\frac{1}{n+1}$ and $\frac1n$.

Or, you could go decimal. There are infinitely many non-repeating strings of digits from $0$ to $9$

Or, take any irrational number $x$ on $(0,1)$ (for example, $x=\frac{1}{\sqrt 2}$. Then, $x,\frac x2,\frac x3,\dots$ are all irrational and all on $(0,1)$.

Alternatively, the set $(0,1)$ is uncountably infinite, while $(0,1)\cap \mathbb Q$ is countably infinite, so the set $(0,1)\cap\mathbb R-\mathbb Q$ must be uncountably infinite as well.

