Is $\mathbb{R}$ dense in $\mathbb{R}$?

I am trying to solve a question from my real analysis book.

Is this statement correct? For every pair x < y of real numbers, there is a real number z such that x < z < y

I was thinking that the statement would be correct if $\mathbb{R}$ were dense in $\mathbb{R}$. (Is that even a correct assumption?)

So that made me think. Is $\mathbb{R}$ dense in $\mathbb{R}$? I am very new to this concept, but I tried to modify a proof that shows that $\mathbb{Q}$ is dense in $\mathbb{R}$.

Proof. Let $x,~y\in \mathbb{R}$ where $x < y$. Let $\displaystyle a = \frac{1}{2}\left( x+y \right)$.$$\frac{1}{2} \left( x + x\right) < \frac{1}{2}\left( x+y \right)<\frac{1}{2} \left( y + y\right)$$$$\iff$$$$x < a < y$$$$\text{Q.E.D}$$Is this proof correct? Does this prove that the original statement is true?