@bershs
Fortunately this isn't on you to prove, our Terms of Service team will have the records at their disposal to take a look at what happened when an action is appealed so don't fret too much in that regard.
You'll just need to make that appeal which you've already done, so from that point it would be a matter of waiting to hear back. That 30 days is the maximum it would typically take, but you should be hearing back substantially faster since we don't have a particularly high volume of those requests currently.
If you want to double-check on a potential payment or account issue feel free to reach out to support directly as well. From that link, the options for Origin | PC | Manage my account | Manage account details would be the best bet to speak with support about the account itself.
Keep in mind support would not be able to review the ban, but should be able to take a look at your recent transactions and account activity to see if there are any other concerns.