Consider the following linear combination
\begin{align*}
1\left(\vect{v}_1-\vect{v}_2\right)+&
1\left(\vect{v}_2-\vect{v}_3\right)+
1\left(\vect{v}_3-\vect{v}_4\right)+\cdots+
1\left(\vect{v}_n-\vect{v}_1\right)\\
&=
\vect{v}_1-\vect{v}_2+
\vect{v}_2-\vect{v}_3+
\vect{v}_3-\vect{v}_4+\dots+
\vect{v}_n-\vect{v}_1\\
&=\vect{v}_1+\zerovector+\zerovector+\cdots+\zerovector-\vect{v}_1\\
&=\zerovector
\end{align*}
This is a nontrivial relation of linear dependence (Definition RLDCV), so by Definition LICV the set is linearly dependent.