In logic, this relation is called almost less than or equal, and is denoted with an asterisks on the relation symbol, like this: $f \leq^* g$.
For example, the bounding number is the size of the smallest family of functions from N to N that is not bounded with respect to this relation. Under CH, the bounding number is the continuum, but it is consistent with the failure of CH that the bounding number is another intermediate value.
