Adding to Vladimir_Nesov’s comment:
In general, every suborder of a well-order is well-ordered. In a word, the property of “being a well-order” is hereditary. (compare: every subset of a finite set is finite)
Adding to Vladimir_Nesov’s comment:
In general, every suborder of a well-order is well-ordered. In a word, the property of “being a well-order” is hereditary. (compare: every subset of a finite set is finite)