I'm looking at the Malliaris-Shelah article about Cofinality Spectrum Problems, and I'm getting some trouble with a result: in Lemma 2.2, they say that as a consequence of Los' theorem, any nonempty bounded definable set of $(\mathbb{N},<)^{\lambda}/\mathcal{D}$ has a greatest and lastest element (with $\lambda$ a regular infinite cardinal and $\mathcal{D}$ a regular ultrafilter). I've been thinking in letting $B \subseteq (\mathbb{N},<)$ nonempty, bounded and definable, so $B$ has greatest and lastest element, and then using an embedding $j:(\mathbb{N},<) \to (\mathbb{N},<)^{\lambda}/\mathcal{D}$ to find a subset $A \subseteq (\mathbb{N},<)^{\lambda}/\mathcal{D}$ with greatest and lastest element. Is this idea useful?
Thanks for your help
Unless I'm misunderstanding, you just need a direct application of Łoś's theorem. If $B$ is a definable set, then there's some first order formula $\phi$ such that $x\in B$ iff $\phi(x)$ is true. Then you can express the idea "$B$ has a greatest element" in a first-order sentence:
$\exists M [\phi(M) \land (\forall x [\phi(x)\implies x \leq M])]$
By Łoś, this statement is true in the ultrapower iff it's true in the naturals.