I am trying to show that $\langle\Bbb Q,<\rangle$ is an elementary submodel of $\langle\Bbb R,<\rangle$.
I first believed that this problem is quite trivial $-$ I thought all I needed to do was show that $\langle\Bbb Q,<\rangle$ and $\langle\Bbb R,<\rangle$ are elementarily equivalent (which follows since both are dense linear orders without endpoint) and then say that $\Bbb Q$ is obviously contained in $\Bbb R$. However, I'm now questioning myself after examining definitions more closely, in particular those related to this post.
In other words, is it not enough to show that the two models are elementarily equivalent, and one a submodel of the other when trying to show one is an elementary submodel of the other?