Let $R$ be a commutative, unital ring. According to wikipedia, a monomial order is a total order $\leq$ on the space of monomials of a given polynomial ring of $R$ such that
If $u\leq v$ and $w$ is some monomial, then $uw\leq vw$;
$1\leq u$ for all $u$.
It is stated that monomial orders are well-orderings. This is not clear at all to me. I googled it but I couldn't find anything proving this. However, there are several places (lecture notes, mostly) where a monomial order is defined as a well-ordering in monomials respecting multiplication.
I am looking for a proof, or at least a reference for this fact?