- #1
Mr Davis 97
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I am reading Abbot's "Understanding Analysis," and in this text he assumes that the real numbers are complete, that is, he assumes the least upper bound property, and begins to prove everything from there. Later in the book he proves that the square root of 2 does in fact exist in ##\mathbb{R}##. I don't understand completely why this proof is necessary. If we assume that the real numbers are a complete ordered field, doesn't that imply that there are no "gaps" and hence that the ##\sqrt{2}## must exist?