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Proof that the irrational numbers are uncountable

We previously proved that $\mathbb{Q}$, the set of rational numbers, is countable and $\mathbb{R}$, the set of real numbers, is uncountable. What can you say, then, about the cardinality of the irrational numbers?

I would say that the cardinality of irrational numbers is uncountable. That would be because the set $\mathbb{Q}$ is a subset of $\mathbb{R}$. The irrationals is also a subset of $\mathbb{R}$. Basically, my thought is that the continuum is composed of irrationals and rationals (only two subsets of reals). Because we can prove that the rationals are countable by Cantor's proof, then the remaining subsets of $\mathbb{R}$, the irrationals, has to be uncountable for $\mathbb{R}$ to be uncountable. My concern is how to go about the prove this (rigorous, intro level).