Proof for INFINITY - Y = INFINITY:-

             INFINITY - Y = INFINITY

suppose Y is not infinity.

lets suppose that          INFINITY - Y =X   ...........................(#)

X will be in our interval and   X< + infinity.
 then we can find  two other elements which are A and B
                    with    A < X < B.

then we can write   A < infinity - Y < B...............from(#).

and we know that Y is also an element of the intervals (since Y is some real no. and not an infinity),,,,
so we can have two other elements  C and D 

                    with  C < Y < D.

      which gives  A+C < ( infinity - Y ) + Y < B+D.

which implies    A+C < infinity < B+D.

and we know that  A, B , C & D are  finite real no.s and not infinite.

so it is ABSURD. ( since infinity is not limited to any real no.s)

so as a conclusion,,,,,,,,

infinity - Y   cant be an element of the interval ,,,,, so it is infinity.

also  X cant be  -infinity because ,,,,in that case,,,,
we will have
                  infinity - Y =  -infinity 
implies      infinity + infinity = Y,,,,,,,,, which is absurd too.

hence infinity - Y = infinity ,,........