The hyperreal number system is defined as one that contains the real numbers, satisfies the first order properties of real numbers, and contain infinitesimals. It can't be as simple as stating the reals are a subset of the hyperreals. Do I need to prove an isomorphism between the two? Or am I completely missing something here?
Any help would be appreciated.
On
The first-order properties of the reals are what we call "real closed field" ... so start with the reals, add an infinite element to that ordered field [say use the rational functions $\mathbb R(t)$ over $\mathbb R$ where $t$ is large and positive]. Then take its real-closure. So it is enough to show that the real-closure of an ordered field of power $\mathfrak c$ still has power $\mathfrak c$. [Concretely, perhaps, a space of Puisieux series.] Maybe this approach has a less "Axiom-Of-Choice" feel than either ultrafilter or Lowenheim-Skolem.
added Dec 14
I am not convinced by André's comment. But anyway, my answer is for the first-order theory of the ordered field of the reals with only these usual relation symbols: $+, \cdot, 0, 1, \lt$. Only the OP can tell us if this is what he really intended.
Since the hyperreals are an extension of the real numbers of course they cannot have a smaller size. You need to show that there is such extension which does not change the size, namely there is a hyperreal field $^\ast\mathbb R$ such that $|^\ast\mathbb R|=|\mathbb R|$.
Hint: Recall that one of the canonical ways to construct such field is using a non-principal ultrafilter $\mathcal U$ over $\mathbb N$, and taking $^\ast\mathbb{R=R^N}/\mathcal U$.