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dalcde

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Is the following proof that the rationals are dense in the reals valid?

Theorem: [tex]\forall x,y\in\mathbb{R}:x<y, \exists p\in\mathbb{Q}: x<p<y[/tex] Viewing x and y as Dedekind cuts (denoting the cuts as x* and y*), x* is a proper subset of y*, hence there exists a rational in x* but not in y*, i.e. there is a rational between x and y.

Theorem: [tex]\forall x,y\in\mathbb{R}:x<y, \exists p\in\mathbb{Q}: x<p<y[/tex] Viewing x and y as Dedekind cuts (denoting the cuts as x* and y*), x* is a proper subset of y*, hence there exists a rational in x* but not in y*, i.e. there is a rational between x and y.

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