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 ,,........
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 ,,........