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I default to the construction of the reals I learned in my first semester Real Analysis course. This is that they are the set of equivalence classes of convergent Cauchy sequences of rationals. I am now wondering where in basic Real Analysis the Axiom of Choice is relied upon. Is that construction of the reals independent of its usage?

If not, can we obtain different models of the real numbers if it is dropped?

(Added) Let me give a specific example of my concern: When I go to prove something like the reals are closed under addition, I say things like: Suppose $x_n$ and $y_n$ are representative cauchy sequences of two reals, then I go on to prove that $x_n + y_n$ is cauchy. Didn't I just use the axiom of choice there? Isn't a choice of representative from each of the equivalence classes (which I made to prove the theorem) a usage?

muaddib
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    AC is not used in the construction. One early place where (a weak form of) AC is used is in showing that sequential convergence is equivalent to convergence. – André Nicolas May 30 '15 at 18:13
  • Thanks Andre. I added something to the question to make it a bit more concrete. – muaddib May 30 '15 at 18:19
  • For future readers: http://math.stackexchange.com/questions/730260/proving-there-is-a-sequence-convergent-to-a-limit-point-of-a-set-without-axiom-o I'll study on that to see how they are different. – muaddib May 30 '15 at 18:37

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The Axiom of Choice is not used in proving closure under addition. We do not need AC to choose one element from a non-empty set.

One early place where (a weak form of) AC is used is in proving that sequential convergence is equivalent to convergence.

André Nicolas
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