Here is a counterexample: $x_1 = \sqrt2, x_2 = 1-\sqrt2$.
For $n>0$, $(nx_1) + (n x_2) =1$ so the limit infimum is $1$, not $0$.

I think the other answers assumed that you meant $x-[x]$, where $[x]$ is the nearest integer to $x$, instead of $(x)$.

If there is no rational dependency, then the multiples are dense in the torus $(\mathbb{R}/\mathbb{Z})^k$, so the limit infimum is $0$. If there is some rational dependency, there will always be small multiples (close to the origin in $(\mathbb{R}/\mathbb{Z})^k$), but these might not be positive.

Consider the closure of multiples of $\langle x_1,...,x_k \rangle$ in the torus $(\mathbb{R}/\mathbb{Z})^k$. This is a closed subgroup, hence a finite union of parallel flat subtori. For $(x)$, the condition you want is that the tangent space to this subgroup at the origin intersects the positive orthant.