As the title suggests, I would like to prove that the normed vector space $(\mathbb{R}^\infty , ||.||_{l^p})$ is not a Banach space, where $\mathbb{R}^\infty :=\{ x:\mathbb{N} \rightarrow \mathbb{R} : \exists \ \bar{n} \in \mathbb{N} \ \ s.t. \ \ x(n)=0 \ \forall n>\bar{n} \}$ and $l^p:=\{x:\mathbb{N} \rightarrow \mathbb{R} : \sum_{n=1}^{\infty}|x(n)|^p< \infty \}.$
As usual, I should start from a Cauchy sequence defined with respect to the distance induced by the considered norm.
Now I'm searching for a sequence of objects in $\mathbb{R}^\infty$ which converges to a sequence in $l^p \setminus \mathbb{R}^\infty$. Can someone give me some hint? I found the completeness of metric and normed spaces a very interesting topic, but I have the impression that the proof of the non-completeness, rather than the completeness, is always harder to achieve. Thank you all.
p.s.: Has the space $\mathbb{R}^\infty$, as defined previously, a particular name?