How would you prove that every graph $G$ is an induced subgraph of the $r$-regular graph, where $r\geq \Delta(G)?$

I can picture the answer for when $G$ itself can be turned into a $\Delta$-regular graph: make a union of $G$ with a copy of itself and then connect the vertices across the two vertex sets $U$ (from $G$) and $W$ (from the copy of $G$) such that $u_i$ and $w_j$ are connected if and only if $v_i$ and $v_j$ would be connected in the original graph in order to turn it (the original graph) into a $\Delta$-regular graph.

However, I cannot figure out how to do it in the general case where, for instance, the order of $G$ may be even or odd (and, thus, may not be made into an $r$-regular graph if r is odd as well) or for when $r>\Delta$. (I am also having trouble with the just language of graph theory and how to write proofs for it if you couldn't tell.)

addresponses to what people have said or asked to the body of the text – Yemon Choi Jan 17 '11 at 19:53