Talk:Real Number Line is Complete Metric Space

From ProofWiki
Jump to navigation Jump to search

I do not think you can use the Bolzano-Weierstrass in "Cauchy sequences converge on the real line" to prove that R^n is complete. After all, Bolzano-Weierstrass requires that R^n is complete in order for the proof to work. So you have a circular argument. -- (unsigned: by Solaris: 12:27, 29 February 2012‎)

At which point does B-W use the fact that $R^n$ is complete? That passed me by. If so, then we need to sort this out. --prime mover 08:09, 29 February 2012 (EST)

The monotone proof uses the existence of a supremum for a bounded set (compare eg. $\Q$ to see that this needs completeness); the exhaustive proof uses directly the completeness. There is (AFAIK) only one way to prove this, and that is to resort to the formal definition of real numbers and show it from there. The definition of $\R$ on PW is de facto that it is the result of the construction given on Completion Theorem (Metric Space). So rather than to reproduce this (lengthy) proof, one could instead just refer to it. When instead the Dedekind cut method was used, the proof would necessarily be different (but it might be a good idea in time to put up more, equivalent, definitions of the real numbers). --Lord_Farin 13:11, 29 February 2012 (EST)

...I'll leave it up to the experts ... --prime mover 15:15, 29 February 2012 (EST)