proving that if graph has perfect elimination order than the graph is perfect
Here's a route you can take.
Prove (or use the fact) that a graph is chordal iff it has a perfect elimination order. Now, it will suffice to show that a chordal graph is perfect. For that, you can take the strong perfect graph theorem stating that a graph is perfect iff it contains no odd hole and no odd antihole. Proceed by contradiction.