Can I share my iMac's internet to iPhone or iPad over USB? I have an iMac, connected to the Internet via Ethernet, and an iPhone 4. I would like to share the iMac's Internet connection to my iPhone via the USB Cable (because my wife dreams of a wifi-less household).

I've set up Internet Sharing, specifying "sharing from: Ethernet  via: USB IPhone", but when I test, the iPhone cannot connect to the Internet. Is there something I need to activate on the iPhone? 

iMac, mac os x version: Mountain Lion, 10.8.4
iPhone 4, IOS Version: 6.1
This works on macOS 10.12 and iOS 10 without needing a jailbreak or other app.
Later macOS 10.13 enhanced this feature as part of asset caching and Apple Configurator 2 features for multiple iOS devices sharing the Mac network. This is out of the box support, built in to the OS with no extra first party apps or modifications needed.
Before iOS 10 and macOS 10.13, USB tethering only worked in one direction or for one device. Before macOS 10.12 - internet service used to flow out of iOS but not in to iOS from a Mac over USB.