Proving your identity is extremely easy. You just need to make a photocopy of your valid USA passport. If your name appears differently on your passport than on your drivers license or other documents, you will need legal documents showing a name change. They are very adamant about this.
Remember, if your passport is not valid for at least 6 more months you will likely have problems. Make sure you renew your passport if it expires in less than 6 months.
Another important thing to note is that you will not get back any of the documents you are submitting. Do not use sentimental heirlooms. Only use official government copies.