A Finite Sum of Vectors with Vanishing Tail

Let $K$ be a field, let $V$ be a vector space over $K$ with zero vector $0_{V}$, let $n$ be a natural number, let $[n]$ be the initial segment it determines, and let $<$ be the strict order on $\mathbb{N}$.

Let $u\in V^{n}$ be an $n$-tuple in $V$ and let $j\in[n]$ be such that $u_{k}=0_{V}$ for every $k\in[n]$ with $j<k$. Then, with finite sums in $V$,