Do what it says, use what you know
A good way to approach proofs is to "Do what it says, use what you know". In other words, look at the structure of what you've already proved (or assumed) and what you are trying to prove.
Often, this can help you make progress. We will present a table of proof techniques in a future lecture, but we list some of the techniques here.
If you know that "P or Q" is true for some statements P and Q, and you wish to show a third statement R, you can do so by separately considering the cases where P is true and where Q is true. If you are able to prove R in either case, then you know that R is necessarily true.
This technique is often referred to as case analysis.
Proving "for all" statements
The fact that arbitrary does not mean you get to pick ; on the contrary, your proof should work no matter what you choose. This means you can't use any property of other than that .is