About 5 years ago I had this working flawlessly working with a Gmail account by the way within the stock Apple mail app. Now you read an e-mail on iCloud.com and you'd expect the notification on your iPhone lock screen and the unread message badge icon on the app to be gone, empty. Nope, not with iOS 10 (still?!). Can't believe this hasn't been fixed yet. Such a basic thing. A minor thing but once you've had it working the obviously better way, going back just feels dumb.