The converse of the previous proposition is not true in general, for instance, the ordered set of integers $(\mathbb Z,\ge)$ is a chain, but it is not well-ordered, since not every of it subsets contains a minimum, e.g. the infinite subset $\{-1,-2,-3,\ldots \}.$ It is only true, if $V$ is finite:

Proposition: Finite Chains are Well-ordered

Every finite chain $V$ is well-ordered.

Proofs: 1

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  1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition