I've been running various gaming communities for 20+ years and if there's one thing I've learned about discipline it's that you should trust your own judgement. Facepunch can't help you make your decisions on your server, that's up to you.
People will constantly demand "proof" from you. You don't have to provide it, and in fact in most cases you SHOULDN'T provide it even if you can, because all it does is give them information they can use to cheat better the next time. It's your community, you decide who has the PRIVILEGE to play on YOUR SERVER. You can ban people for no reason at all if you like, you don't owe anyone explanations.
Always bear in mind that your responsibility is to the well behaved players who you want to keep on your server - these people will respect you for taking a hard line against undesirables, for whatever reasons you see fit. In the rare cases that they think you made a mistake you'll be able to have a calm, rational discussion about it with them.
As for the opinions of the banned players, who cares? F*** 'em!
as for the ban you banned without any proof what so ever i don' t blame them calling you trash.
Trash calling me trash has never bothered me, it just makes me glad I got rid of them in the first place.