Learn Build an ordered complete field from rational Cauchy sequences, proving arithmetic, order and completeness without assuming real limits in advance.
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Different rational sequences can represent the same new number.
Equivalent representatives must produce equivalent outputs; a formula on representatives is not automatically well-defined on classes.
This unit constructs limits rather than assuming they already exist.
No. Its definition only compares sufficiently late terms using each positive rational tolerance. Do not invoke completeness of the real numbers as a premise.
Review the Cauchy definition; set aside the real completeness theorem
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Representatives, diagonals and a new irrational cut
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Build field operations on equivalence classes of rational Cauchy sequences using rational estimates, including a justified reciprocal construction.
6 practice tasks.
Construct an ordered field from rational Cauchy classes without borrowing real completeness, then prove rational approximation and density.
6 practice tasks.
Build limits from rational approximations, derive least upper bounds, and construct an irrational root without assuming it already exists.
6 practice tasks.