Exercise 2.4.3 (Challenging): Suppose F is an ordered field that contains the rational numbers, such that Qis dense, that is: Whenever x, y = F are such that x < y, then there exists a q EQ such that x 0, there exists an M such limit - that for all n,k> M, we have |xn − xk| <ɛ. Suppose every Cauchy sequence of rational numbers has a in F. Prove that F has the least-upper-bound property.

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Exercise 2.4.3 (Challenging): Suppose F is an ordered field that contains the rational numbers Q, such that
Qis dense, that is: Whenever x,y F are such that x <y, then there exists a q EQ such that x<q <y. Say
a sequence {x}=1 of rational numbers is Cauchy if given every & € Q with & > 0, there exists an M such
that for all n,k> M, we have |xn− xk|< E. Suppose every Cauchy sequence of rational numbers has a limit
in F. Prove that F has the least-upper-bound property.
Transcribed Image Text:Exercise 2.4.3 (Challenging): Suppose F is an ordered field that contains the rational numbers Q, such that Qis dense, that is: Whenever x,y F are such that x <y, then there exists a q EQ such that x<q <y. Say a sequence {x}=1 of rational numbers is Cauchy if given every & € Q with & > 0, there exists an M such that for all n,k> M, we have |xn− xk|< E. Suppose every Cauchy sequence of rational numbers has a limit in F. Prove that F has the least-upper-bound property.
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