Bringing The World Together By Tunnels

When i was looking into if it was possible to circumnavigate the globe by train, i noticed one big problem, Oceans.
Once you get to Japan there is the Pacific bringing you to a halt and once you overcome that and get to the West coast of America, there is the Atlantic in the way.
Approximately 3000 miles of wet, blue stuff lays between the USA and Britain and the only way across is an expensive trip aboard a cruise-liner that runs infrequently.
As a bridge between the UK and America is out of the question, what about a tunnel?
It wouldn't even need to go from the UK to USA, it could go via Iceland and Greenland and end in Canada so you have three 1000 mile tunnels instead of having to dig one big long one. 
Stick in a train track and North America and Europe are within 12 hours of each other. Dig another one in the Bering Strait between Alaska and Russia and that's the whole World joined together, well the important bits, Australia would have to stay stuck out on it's own i'm afraid.
It would be good for the environment as less flights needed across the Atlantic, good for the economy as more people would travel between the continents and most importantly i could then travel around the World by train and people would get to go to Greenland because as far as i know, nobody ever goes to Greenland.
It's a plan just waiting to happen so Cameron, Obama, Putin stop arguing over how to bomb Syria and hire a really big digger and let's go, the people of Greenland are waiting.