Show there is a sequence of rational numbers converging to

tapiat0wa4c 2022-06-03 Answered
Show there is a sequence of rational numbers converging to any irrational number.
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traforatp5il7
Answered 2022-06-04 Author has 1 answers
The space of real numbers R is the metric completion of Q the space of rational numbers. To construct R, start with Q, and let C be the space of Cauchy sequences of Q. Two elements ( x n ) and ( y n ) are equivalent if and only if ( x n y n ) converges towards 0, then R is the quotient space for this equivalence relation. You can identify Q with the equivalence classes of constant sequences.

If [ x n ] is the class of the Cauchy sequence ( x n ), define ( y n i ) by y l i = x i for every l. Then ( [ y n i ] ) converges towards [ x n ] and [ y n i ] can be identified with the rational x i .
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