The game is (restating definition) that for every epsilon, there exists an N such that every n,m > N, then the difference of the sequence terms is smaller than epsilon.
The sqrt sequence breaks this as for any N, we an choose n=N and m=4N=4n (here were being a bit lax in that n=N but that's not important). The difference is sqrt(n)=sqrt(N). This will not be smaller than any given epsilon as we can make the difference as large as necessary by choosing N sufficiently large.
The sqrt sequence tends to infinity although the difference between successive terms goes to zero. Choosing m=4n could have been m=9n or ... and the difference can be made as large as you want for any N.