The problem with all such schemes is that 'the internet' does not exist as an entity. There is a reason why failed connection attempts are reported as 'unreachable' or 'cannot connect to server at blahblah'. Examples:
1) You have no signal. Are you connected to the internet? Will PING succeed? Can you connect to your target server?
2) You have a signal, but your provider data allowance has been exceeded. Are you connected to the internet? Will PING succeed? Can you connect to your target server?
3) Your provider connection is fine, but their backbone router is down. Are you connected to the internet? Will PING succeed? Can you connect to your target server?
4) Your provider connection is fine, their backbone router is up but the fibre connection to country X where the server is has been interrupted by some drunken Captain and his ship's anchor. Are you connected to the internet? Will PING succeed? Can you connect to your target server?
5) All the links to the target country are up but Fred, with his ditch-digging JCB, has cut the power cable to the server farm. One of Fred's other jobs is to service the backup generator:( Are you connected to the internet? Will PING succeed? Can you connect to your target server?
6) All the hardware is up, but the server code was written by Fred before he was demoted to ditch-digger for incompetence and has now crashed, again. Are you connected to the internet? Will PING succeed? Can you connect to your target server?
7) Fred has had a day off, but his replacement, competent server admin has blocked off ICMP ping in the routers to prevent ping DOS attacks. Are you connected to the internet? Will PING succeed? Can you connect to your target server?
So, the ony way to be sure is to attempt to connect to the target server and see what happens.
You can surely detect some negative cases more quickly - surely if there is no signal, you cannot get a connection:) Past that, you should just try to connect. Tell the user what is going on, use a timeout and supply the user with a 'Cancel' button. That's about the best you can do.