Union of a Family of Nested Ideals

Here, we have a family of ideals {I_j} in a ring R. Here j runs through some index set say A.

Also I_1 >= I_2 >= I_3 >= ...

Let's try to show that I = UNION(I_j, j in A) is an ideal.

I is nonempty since each I_j contains 0.
Take r in R and x in I. Then x is in some I_j, so rx is in I_j because it's an ideal so rx is in I.
Take x in I, so x lives in some I_j. Then -x is in I_j so it's also in I, so I is closed under inverses.

Now the more interesting part.
Take x, y in I. Then x is in some I_j and y is in some I_k.
If j = k then x + y is in I_j and hence in I and we are done.
Assume wlog that j > k. Then I_j <= I_k, so x is in I_k. Thus both x and y are in I_k hence so is x + y, and so x+y is in I as well, done.