Should I Trust the Chat Confirmation?

I cancelled my broadband with 33 days notice. An agent in chat confirmed everything was in order and said I would get an email about the return kit and other details. Three days later, I hadn’t received anything, so I chatted again. The second agent confirmed the cancellation but said there’s no email and that the chat transcript is enough. They also confirmed I will be disconnected on the requested date. Has anyone else had this experience? 

Should I just leave it and trust that the disconnection will happen as promised? Or should I send a formal letter or do something else to make sure it’s properly processed?