There have been several classical constructions of the reals that avoid these problems, the most famous ones being Dedekind Cuts and Cauchy Sequences, named respectively for the mathematicians Richard Dedekind (1831 - 1916) and Augustine Cauchy (1789 - 1857). We will not discuss these constructions here, but will use a more modern one developed by Gabriel Stolzenberg, based on interval arithmetic.

Steam Engines. […] The valve rods are coupled by connecting-rods […] and yokes[…], to eccentrics[…].

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The cocktail was a heady mixture of spirits.