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math771
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olarBear: Yes, all of these formulations of completeness are equivalent. It is a fundamental property of real numbers and metric spaces, so it makes sense that any formulation of completeness would be equivalent to the others.
To answer your specific questions, 3 does imply 1, and 5 and 6 do become equivalent statements when considering ℝ as a metric space with the standard Euclidean metric. This is because the standard Euclidean metric satisfies the conditions for a metric space to have a Cauchy completeness and nested interval property.
I hope this helps! Good luck with your proofs.
To answer your specific questions, 3 does imply 1, and 5 and 6 do become equivalent statements when considering ℝ as a metric space with the standard Euclidean metric. This is because the standard Euclidean metric satisfies the conditions for a metric space to have a Cauchy completeness and nested interval property.
I hope this helps! Good luck with your proofs.