= Density of the irrational numbers
{title2=$\overline{\mathbb R\setminus\mathbb Q}=\mathbb R$}
Every nonempty open real interval contains an <irrational number>. Choose a <rational number> $q$ strictly inside the interval using <density of the rational numbers>; then $q+\sqrt2/N$ remains inside for sufficiently large positive <integers> $N$ and is irrational. Thus the irrational numbers form a <dense subset> of the real line. Together with rational density, this gives approaching <sequences> of both types at every real point, useful for testing <continuity> of functions defined differently on the two sets.
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