Directly use the Bolzano-Weierstrass theorem (Theorem 2.3.8) to prove that every Cauchy sequence of real numbers...

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Directly use the Bolzano-Weierstrass theorem (Theorem 2.3.8) to prove that every Cauchy sequence of real numbers is convergent. That is, only make use of the fact that every bounded sequence of real numbers has some convergent subsequence (not necessarily converging to either lim sup/lim inf). (Remark: Proving Cauchy-completeness from Bolzano-Weierstrass without lim sup/inf in this way generalizes more easily to R".) (b) One interesting thing about the Cauchy sequence definition that it can be stated without reference to real numbers at all: We say a sequence of rational numbers {rn} is Cauchy (in the absolute value metric on the rational numbers) if for every & EQ satisfying > 0, there exists some MEN such that for all n, k M, we have rn - rk | Directly use the Bolzano-Weierstrass theorem (Theorem 2.3.8) to prove that every Cauchy sequence of real numbers is convergent. That is, only make use of the fact that every bounded sequence of real numbers has some convergent subsequence (not necessarily converging to either lim sup/lim inf). (Remark: Proving Cauchy-completeness from Bolzano-Weierstrass without lim sup/inf in this way generalizes more easily to R".) (b) One interesting thing about the Cauchy sequence definition that it can be stated without reference to real numbers at all: We say a sequence of rational numbers {rn} is Cauchy (in the absolute value metric on the rational numbers) if for every & EQ satisfying > 0, there exists some MEN such that for all n, k M, we have rn - rk |

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