In the construction of real numbers from cauchy sequence, we define real number to be the set of equivalence classes of cauchy sequences of rational numbers. But how can we ensure that all cauchy sequences of rational numbers converges (in Q or in R), what if there exists a cauchy sequence of rational number that diverges? A divergent cauchy sequence of rational would still represent a real number by our definition, but what kind of “number” can a divergent sequence represent? Or is this some assumption we all have to agree on?
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$\begingroup$ Who assumes that all Cauchy sequences of rational numbers converges? $\endgroup$– José Carlos SantosJan 1 at 22:01
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$\begingroup$ They don't. We identify the Cauchy sequences with the real numbers. $\endgroup$– John DoumaJan 1 at 22:01
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3$\begingroup$ Cauchy sequences of rationals (or rather, equivalence classes of these) are defined to be real numbers. Hence they all converge in $\mathbb R$ as a matter of definition. Of course, one does have to prove that this construction makes sense, and that's not an easy business. $\endgroup$– luluJan 1 at 22:06
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$\begingroup$ Should add: other constructions of $\mathbb R$ are possible and if you are working with one of those, you can demonstrate the convergence as a theorem about that system. But you'd need to specify how you are defining $\mathbb R$ to say much more. $\endgroup$– luluJan 1 at 22:11
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$\begingroup$ What's your mental image of a "divergent" Cauchy sequence? Are you sure the thing you're picturing is Cauchy? $\endgroup$– KarlJan 1 at 22:48
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1 Answer
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As stated in the comments, strictly speaking, your question only really makes sense if you have already constructed the real numbers in some other way. Otherwise, we are just defining a real number to be a Cauchy sequence of rational numbers, so they converge by definition.
But it is still reasonable to ask why we want to define a real number to be a Cauchy sequence of rational numbers. This is because the real numbers are meant to model a certain geometric object - a line. Now if you picture a Cauchy sequence of rational numbers in a line (we put rational numbers on a line in the obvious way), then geometrically it is quite plausible that there is an actual point on the line which is the limit of those rational numbers. That's the reason for constructing real numbers as Cauchy sequences of rational numbers.
