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Proof That the Real Numbers Are a Completion of the Set of Rational Numbers

Theorem

The real numbersare a completion of the set of rational numbers

Proof

A metric spaceis a completion of a metric sp[aceifis complete and is isometric to a dense subset ofso thatfor

The closure ofinissinceis dense in

is complete with the Euclidean metric so with the Euclidean metric onandis a completion of