Page "Construction of the real numbers" Paragraph 42

This does indeed define an equivalence relation, it is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all the usual axioms of the real numbers.

This is remarkable because not all of these axioms necessarily apply to the rational numbers, which are being used to construct the sequences themselves.

We can embed the rational numbers into the reals by identifying the rational number r with the equivalence class of the sequence ( r, r, r, …).