TITLE: Is a model of arithmetic contained in a model of arithmetic an initial segment?
QUESTION [9 upvotes]: It's easy enough to show that if $\mathbb{N}_1$ is a non-standard model of the Peano axioms, then there is a canonical embedding $\mathbb{N} \to \mathbb{N}_1$, and we have a theorem that if $x \in \mathbb{N}_1$ and $y \in \mathbb{N}$ such that $x < y$, then $x \in \mathbb{N}$.
What if we had two non-standard models $\mathbb{N}_1 \subseteq \mathbb{N}_2$, ideally an elementary embedding? Must it be true that if $x \in \mathbb{N}_2$ and $y \in \mathbb{N}_1$ such that $x < y$, then $x \in \mathbb{N}_1$? 
I'm also curious about the analogous questions for models of real analysis or of ZFC; e.g. comparing the set of integers in two nonstandard models of analysis.

REPLY [3 votes]: Andres Caicedo has answered the question as stated, however let me point out that one can recover a form of the property under further assumptions on the pair of models. Namely, let $\mathbb N_1$ be a model of PA, and $\mathbb N_2$ a model of Robinson arithmetic which is definable in $\mathbb N_1$: that is, there is a definable set $D\subseteq\mathbb N_1$ and definable functions $\oplus,\odot\colon D^2\to D$ such that $\langle D,\oplus,\odot\rangle$ is (isomorphic to) $\mathbb N_2$. Then there exists a (unique) definable embedding of $\mathbb N_1$ onto an initial segment of $\mathbb N_2$.