Answer to Question #122941 in Real Analysis for Ruksan

Question #122941
Use what you know from analysis on R to come up with a definition of a
Cauchy sequence in V . When would you say V is complete? Is Rn with
|| . ||∞ complete?
1
Expert's answer
2020-06-24T19:20:57-0400

A sequence "\\{x_n\\}" in "\\R" is called Cauchy sequence if

"\\forall \\ \\epsilon, \\exist \\ M \\in \\N : |x_n-x_m|<\\epsilon \\ for \\ n,m\\geq M" .

We know Every convergent sequence is a Cauchy sequence.


The space M is called complete if every Cauchy sequence in M converges to a point in M.


"R^n" with || . ||∞ is not complete because there exists a Cauchy sequence in "R^n" which does not converge to a point in "R^n" .





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