How can I use a Macbook Air to provide a TV with an internet connection?
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.
If you have the USB Ethernet adapter, you should be able to follow the instructions found here:

Set up the Sharing -The first step is to set up sharing on the host Mac. Head to System Preferences &gt; Sharing
Set up the client - make sure your TV uses DHCP

General instructions from Apple support:

To set up Internet sharing:

Choose Apple menu &gt; 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.