Greatest Element of a Finite Family in a Totally Ordered Set

Let $A$ be a set equipped with a total order $\le$, let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $c\in A^{n}$ be an $n$-tuple in $A$, with components $c_{k}$.

Then there exists $j\in[n]$ such that $c_{k}\le c_{j}$ for every $k\in[n]$.