THE EXISTENCE OF SQUARE ROOT OF TWO

The importance of the Supremum Property lies in the fact that it guarantees the existenceof real numbers under certain hypotheses. We shall make use of it in this way many times. At the moment, we shall illustrate this use by proving the existence of a positive real number x such that x 2 = 2; that is, the positive square root of 2. It was shown earlier that such an x canot be a rational number; thus, we will be deriving the existence of at least one irrational number.

Theorem.
There exists a positive real number x such that x 2 = 2

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