I have just finished reading the proof of Downward Lowenheim-Skolem Theorem and have been reading the article Eventually constant variable assignments and the second edition of the book, “A Friendly Introduction to Mathematic Logic”
However, I still don’t know what’s the use of eventually constant variable assignment in the proof of Downward Lowenheim-Skolem Theorem I understand the definition of eventually constant variable assignment as being $\forall i, j, \exists k \in \mathbb{N}, i, j > k, s’(v_{i}) = s’(v_{j})$
Why it must be eventually constant?
