The convergence of unordered sum

$X$ is an arbitrary normed vector space. $A=\{x_i\in X\mid i\in J\}$ is an indexed set. $J$ contains the indices and is uncountably infinite. Let $\mathcal F=\{F\mid F\subseteq J, F \text{ is finite}\}$.

The following definition of the convergence of unordered sum is taken from https://www.math.ucdavis.edu/~hunter/m201b_old/sums.pdf. We claim that the unordered sum $\sum \limits_{i\in J}x_i$ converges if and only if $$\exists x\in X,\ \forall \varepsilon>0,\ \exists F_\varepsilon \in \mathcal F,\ \forall F \subseteq F_\varepsilon \in \mathcal F,\ \left\|\sum_{i\in F} x_i-x\right\|\leq \varepsilon.$$ Suppose the unordered sum $\sum \limits_{i\in J} x_i$ converges. Let $I\subseteq J$. Does $\sum \limits_{i\in I}x_i$ converge? (Note that it may converge to a different point) How to prove it? Thanks!