Prove that the greatest integer function: $\mathbb{R} \rightarrow \mathbb{Z}$ is onto but not $1-1$ Statement: the greatest integer function int: $\mathbb{R} \rightarrow \mathbb{Z}$ is onto but not $1-1$
Proof: let $x \in \mathbb{R}$, then $int(x) \leq x$ and is an element of $\mathbb{Z}$. Since $\mathbb{Z}$ is an element of $\mathbb{R}$, the greatest integer function maps onto $\mathbb{Z}$. However, it is not one-to-one, because $int(0.2)=int(0.3)=0$
Is my proof valid? 

You'd want to more clearly show how given $x \in \mathbb{Z}$ you could find a real number $r \in \mathbb{R}$ such that $\text{int}(r)=x$. Then you have onto. As for $1:1$, you have shown that it cannot be $1:1$. Note that a function that can be plotted in $\mathbb{R}^2$ can only be $1:1$ if every horizontal line drawn in the plane intersects the graph at most one point--clearly not the case here.
