Triangle Inequality for Finite Sums of Vectors

Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$, let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $v:[n]\to V$ be a map with values $v_{k}$. Then

where the sum on the left is the finite sum in $V$, the sum on the right is the finite sum in the field of real numbers, and the order is that of the ordered field of real numbers.