working on proving:
Let $x,y \in \mathbb{R}$ be such that $x < y$. There exists $z \in \mathbb{R}$ such that $x < z < y$
This seems obvious but I'm having trouble using the properties of the real numbers to get this proof started in a way where it will be succinct. Any help is appreciated.
As good a point is any as the midpoint between $x,y$, i.e. $(x+y)/2$. Any scalar multiple (between zero and one) of $x+y$ will do but the midpoint is always the easiest to think over.
So, we suppose
$$x < \frac{x+y}{2} < y$$
and want to prove this. To show the first half of the inequality, notice that, since $x<y$, $x/2 < y/2$. As a result,
$$\frac{x+y}{2} = \frac x 2 + \color{blue}{\frac y 2} > \frac x 2 + \color{blue}{\frac x 2} = x$$
Using the same property - $x/2 < y/2$ - you can similarly get $(x+y)/2 < y$. I'll leave that bit to you.
I suppose if you want to rigorously complete the proof you also have to show that $(x+y)/2$ exists and is a real number in the first place, but we immediately get that since $x,y,2\in \Bbb R$ and the real numbers form a field. Not sure if that level of rigor would be expected in your analysis class though.