Find the Completion of (X, d) where $X = \{\{x_n\} : \sum_{n=1}^\infty 2^n \vert{x_n}\vert < \infty\}$ and $d(x,y) = sup_n \vert x_n - y_n \vert$

Question

I have showed that sequence such that $x^{(k)}_n = (\frac{1}{2 + \frac{1}{k}})^n$ is Cauchy but it converges to $ \frac{1}{2^n}$ which is not in X. But I was not able to find its completion. How can I do this?

Answer 1

the elements of your space are sequences converging to zero, satisfying certain condition on their growth. Now the basic vectors e_n(sequence having all terms zero, except the n-th equal 1) belong to your space, so does their (finite) linear combinations(Since it is a vector space), these are dense in $C_{0}$ (space of sequences converging to zero endowed with the same distance as your space), Since the completion of a metric Space is unique up to isometry, we conclude that the completion of your space is $c_{0}$(of course $c_{0}$ is complete.

Answer 2

The completion of $X$ is the space $c_0$ of sequences which tend to $0$. If $(x_n) \in X$ then $2^{n}x_n \to 0$ which implies $x_n \to 0$. Of course, $c_0$ is a complete space. It remains to shot that $X$ is dense in $c_0$. Let $x_n \to 0$. Then $(x_1, x_2,..,x_N,0,0,..)$ ($N=1,2,...)$) is as sequence on $X$ which converges to $(x_n)$.