The Density of the Irrational Numbers

by Corry Shores
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Scott Wollschleger asks in the comments to the The Density of the Rational Numbers post:

what about the density of an irrational number?

In this previous post we said that between any two rational numbers a and b there is another rational number c. We may make each of these irrational by setting them over the square root of 2.

Thus between any two irrational numbers there is another irrational number: the density of the irrationals.

based on the proof in:

http://math.nyu.edu/~tsang/classes/precalc1/session1.pdf