I am trying to install Real Player. However it keeps telling me that I must be connected to the internet to install. Of course I AM ALREADY connected!! Any thoughts, please?

Check your firewall to see if it's denying access to Real Player,.. if it is, simply "enable" access for it.

Try Real Alternative 2.0.2; I would not have Real Player on any of my machines. I am sure you can find it yourself but here is a link:
Free-Codecs.com :: Download Real Alternative 2.0.2 : Real Alternative will allow you to play RealMedia files without having to install RealPlayer/RealOne Player

I think this problems comes due to your firewall or Antivirus setting. After proper setup try it again.