Let $X = \left\{(x_n):\sum\limits_{n=1}^\infty n|x_n|<\infty\right\}$, $d(x,y) = \sup|x_n - y_n|$. Show that the space $(X,d)$ is not complete. Prove that the space $c_0$ is its completion.

For the solution, I know that a metric space $(X,d)$ is complete if every Cauchy sequence $(x_n)$ in $X$ converges to a point in $X$ so for the first part I need to find a Cauchy sequence that does not converges to a point in $X$.

For $(x_n) = (1/n^3)_{n=1}^\infty$ and $(y_n) = (1/n^4)_{n=1}^\infty,$ $d(x,y)$ not converges. Thus $X$ is not complete. Is my counter example right?

For the second part, proving $c_o$ is its completion I am having trouble about what kind of approach I should follow.

  • $\begingroup$ you miss understand that sequence in $X$ is in fact a sequence of sequences (confusing right?). So you want to find a sequence of sequences that does not convergence to a sequence in $X$ (try to find a sequence (of sequences) that converge to the sequence $1/n^2$.) $\endgroup$ – Yanko Oct 30 '17 at 21:47
  • $\begingroup$ @yanko I think it is about my typo. $x_n$ = $(1/n^3,1/n^3,1/n^3,..)$. Yes that is what I want to find. $\endgroup$ – Pumpkin Oct 30 '17 at 21:50
  • $\begingroup$ Each $x_n$ is a constant sequence and is therefore not contained in $X$. $\endgroup$ – amsmath Oct 30 '17 at 21:52
  • $\begingroup$ Right I get it now, so you should carefully read amsmath's comment. Also I would consider sequences that turns out to be 0 at some point. $\endgroup$ – Yanko Oct 30 '17 at 21:53
  • $\begingroup$ @amsmath Sorry for the typo I corrected my $x_n$ example now it is not constant? $\endgroup$ – Pumpkin Oct 30 '17 at 21:54

For the first Q: Note that for if $x_n=0$ for all but finitely many $n$ then $(x_n)_n\in X.$

Let $x[j]=(x_{j,n})_n$ where $x_{j,n}=1/n$ for $n\leq j$ and $x_{j,n}=0$ for $n>j.$

Then $(x[j])_j$ is a Cauchy sequence in $X$. If this sequence converged to $y=(y_n)_n\in X$, then it would be necessary that $y_n=1/n$ for every $n.$ But $(1/n)_n\not\in X.$


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