Consider the system $R$ of all subsets of the real line containing only finitely many points, and let the measure $\mu(A)$ of a set $A \in R$ be defined as the number of points in $A$. Prove that:
1) $R$ is a ring without a unit
2) $\mu$ is not sigma-additive.
I believe I have answered Part 1, but do not know how to proceed with part 2.
So for 1) I have said if we consider sets $A, B \in R$ then they have a finite number of points and hence $A \Delta B$ and $A - B$ will also have finite number of points and hence $A \Delta B, A-B \in R$ so it a ring. The big triangle means symmetric difference. Suppose the ring has a unit $x$. We know that the sets of {1}, {2}, {3}, .... are in the ring because they are finite subset of the real line. And if $x$ is indeed the unit then these sets will be contained in $x$ hence the positive integers are a subset of $x$ which is a contradiction since any element of the ring must have a finite number of elements. So there is no unit.
2) I'm not sure how to proceed since we are dealing with a ring which means it is closed upto finite union and so I'm not sure if I am supposed to find a counterexample or what.
It would be greatly appreciated if you could check if my part 1 is correct and help me with part 2.