The television we have supports the use of the internet through an Ethernet cable or wireless USB network adapter (bought separately). The television has no built-in wireless and the nearest ethernet port in the house is too far away to be practical.
You should be able to do this via Ethernet or wireless, depending on whether you want to buy a wireless dongle for your TV or whether you have the Ethernet adapter for your MacBook Air.
- Set up the Sharing -The first step is to set up sharing on the host Mac. Head to
System Preferences > Sharing
- Set up the client - make sure your TV uses DHCP
General instructions from Apple support:
To set up Internet sharing:
Choose Apple menu > System Preferences and click Sharing.
Select Internet Sharing.
Choose the Internet connection you want to share from the “Share your connection from” pop-up menu. For example, if you’re connected to the Internet over Ethernet, choose Ethernet.
Select how you want to share your Internet connection in the “To computers using” list. For example, if you want to share your Internet connection over AirPort, select AirPort.
If you share your Internet connection using AirPort, click AirPort Options and give your network a name and password.
Either approach should work - it's most likely a matter of the hardware that you have (or are willing to acquire). Getting the USB Ethernet adapter for your MacBook Air is likely to be cheaper (and of more use long-term) than a dongle for the TV.