Exercise 1.4.5

Question

Using Exercise 1.4.1, supply a proof that $\mathbf{I}$ is dense in $\mathbf{R}$ by considering the real numbers $a-\sqrt{2}$ and $b-\sqrt{2}$. In other words show for every two real numbers $a<b$ there exists an irrational number $t$ with $a<t<b$.

Ulisse Mini · Accepted Answer

The density theorem lets us find a rational number $r$ with $a-\sqrt2 < r < b-\sqrt2$, adding $\sqrt 2$ to both sides gives $a < r +\sqrt 2< b$. From 1.4.1 we know $r+\sqrt2$ is irrational, so setting $t = r+\sqrt2$ gives $a<t<b$ as desired.

Exercise 1.4.5

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Exercise 1.4.5

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