Let $X=\{\{x_n\}:\sum_{n=1}^{\infty}|x_n|<\infty\}$ be the vector space consisting of all real-valued sequences that are absolutely summing. Define \begin{align}P_k:X\to \Bbb{R}\end{align} \begin{align}\{x_n\}\mapsto P_k(\{x_n\})=x_k,\;\;\forall \,k\in \Bbb{N}\end{align} I want to construct a sequence of elements in $X$ such that $\lim\limits_{i\to \infty} P_k (x^{(i)} )$ exists for all $k \in\Bbb{N}$, but such that the sequence $\{x^{(i)}\}$ itself fails to be a Cauchy sequence. The norm on $X$ is given by $\Vert \{x_n\}\Vert=\sum_{n=1}^{\infty}|x_n|$.
MY THOUGHT
I'm thinking of $x^{(i)}=\frac{(-1)^i}{i},\;\forall \,i\in \Bbb{N}$. Am I right?