Consider the normed space $(X, \Vert \cdot\Vert) $ where
$$
X=\{ (a_n)_n \quad|\quad (a_n)_n \text{ real sequence with } \lim_{n\to \infty}a_n=0 \}
$$
and
$$\Vert (a_n)_n\Vert:= \sum_{n\geq 1}\frac{|a_n|}{2^n} .$$
I am looking for a Cauchy sequence in $(X, \Vert \cdot\Vert)$ which does not converge.
Thanks
Best Answer
For each $m$ define a sequence $(a^m_n)_n$ by $$a^m_n=\begin{cases}1;&n\leq m\\ 0;&n>m\end{cases}$$ Clearly each $(a^m_n)_n$ is in $X$. Furthermore, if $m\geq m'$ then $$\|(a^m_n)_n-(a^{m'}_n)_n\|=\sum_{k=m'+1}^m\frac{1}{2^k}\leq \frac{1}{2^{m'}}$$ which shows that the sequence of the $(a^m_n)_n$ is Cauchy. But this sequence cannot converge. Any potential limit $(b_n)_n$ in $X$ has to have $\lim_n b_n=0$, so in particular $b_n\leq \frac{1}{2}$ for all $n\geq N$ for some large enough $N$. But then $$\|(b_n)_n-(a^m_n)_n\|\geq \frac{m-N+1}{2}$$ for any $m\geq N$, showing that the $(a^m_n)_n$ do not converge to $(b_n)_n$.