Question

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?

Answer 1

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 $R^n$ .