Given a Banach vector space $X$, I need to consider an infinite tensor product of it. The simplest construction I found, in Guichardet, is by considering the inductive limit of the direct system $(X^{\otimes i}, \phi_{ij})_{i,j}$ defined by: $$ \begin{array}{lccc}\phi_{i,j}:& X^{\otimes i} &\rightarrow& X^{\otimes j}\\ &\bigotimes_{k=1}^i x_k &\mapsto & \left(\bigotimes_{k=1}^i x_k\right)\otimes e^{\otimes j-i} \end{array} $$ for $i\leq j$, where $e\in X$ is such that $\Vert e\Vert=1$ and for every $i$, $X^{\otimes i}$ is endowed with a crossnorm.
With this, the inductive limit $X^{\otimes \mathbb{N}}$ is the vector space spanned by $\{\bigotimes_{i=1}^\infty x_i~|~x_i=e \text{ for all but finitely many } i\}$ and the canonical mappings are:
$$ \begin{array}{lccc}\phi_{i}:& X^{\otimes i} &\rightarrow& X^{\otimes \mathbb{N}}\\ &\bigotimes_{k=1}^i x_k &\mapsto & \left(\bigotimes_{k=1}^i x_k\right)\otimes \left(\bigotimes_{n=1}^\infty e\right) \end{array} $$
We can put on $X^{\otimes \mathbb{N}}$ the inductive limit norm, which is well defined: given an element $\bar{x}=\bigotimes_{i=1}^\infty x_i\in X^{\otimes \mathbb{N}}$, there exists $m$ such that $\bar{x}=\phi_m\left(\bigotimes_{i=1}^m x_i\right)$, and it comes $\Vert \bigotimes_{i=1}^\infty x_i \Vert = \prod_{i=1}^m \Vert x_i\Vert$.
Then I want to consider the completion of $X^{\otimes \mathbb{N}}$ with respect to this norm. Let's note it $\hat{X}^{\otimes \mathbb{N}}$. In Guichardet (https://www.fuw.edu.pl/~kostecki/scans/guichardet1969pt2.pdf), they only consider a family $(x_i)_i$ of vectors in $X$ such that $\sum_{i=1}^\infty \Vert x_i - e\Vert<\infty$. It is proved that in this case the product $ \prod_i \Vert x_i\Vert $ exists, it is null iff one of the $x_i$ is null, and the family of vectors $\left(\phi_n\left(\bigotimes_{i=1}^n x_i\right)\right)_n$ has a limit in $\hat{X}^{\otimes \mathbb{N}}$, whose norm is given by $\prod_i \Vert x_i\Vert $.
My question is the following: what about the case where $\prod_i \Vert x_i\Vert$ diverges to $0$ ?
For instance let's take $(x_i)_i$ such that $0 < \Vert x_i\Vert \leq \dfrac{1}{N}$ for some $N>1$.
Then $\left(\phi_n\left(\bigotimes_{i=1}^n x_i\right)\right)_n$ is Cauchy. Indeed: $$\forall \epsilon>0, \exists k \text{ s.t. } \dfrac{1}{N^k}<\dfrac{\epsilon}{2}, \forall n\geq m\geq k: \quad \left\Vert \phi_m\left(\bigotimes_{i=1}^m x_i\right) - \phi_n\left(\bigotimes_{i=1}^n x_i\right)\right\Vert\begin{aligned}[t] &=\left\Vert \bigotimes_{i=1}^m x_i \otimes\left(\bigotimes_{i = m+1}^n e-\bigotimes_{i=m+1}^n x_i\right)\otimes \bigotimes_{i=n+1}^\infty e\right\Vert\\ &= \prod_{i=1}^m \Vert x_i\Vert\cdot \left\Vert\bigotimes_{i = m+1}^n e-\bigotimes_{i=m+1}^n x_i\right\Vert\\ &\leq \prod_{i=1}^m \Vert x_i\Vert\cdot \left(\left\Vert\bigotimes_{i = m+1}^n e\right\Vert + \left\Vert\bigotimes_{i=m+1}^n x_i\right\Vert\right)\\ &\leq \prod_{i=1}^m \Vert x_i\Vert\cdot \left(1 + \prod_{i=m+1}^n \Vert x_i\Vert\right)\\ &\leq 2 \prod_{i=1}^m\Vert x_i\Vert <\epsilon. \end{aligned}$$
So the completion should give that $\lim \phi_n\left(\bigotimes_{i=1}^n x_i\right)= \bigotimes_{i=1}^\infty x_i\in \hat{X}^{\otimes \mathbb{N}}$, having norm $0$ ? Consequently it should be a representation of the null vector in $\hat{X}^{\otimes \mathbb{N}}$.
I'm extremely unsure of this conclusion, so if anyone have insights about why my reasoning is false, or could confirm me if it is indeed right, I will be more than grateful.
Thank you in advance for your help.