Hello everybody,
I was wondering if this infinite intersections of sets was empty.
I have sets for all . I was wondering if 0 was in the set of the intersection of all these subsets. I can prove that all negative or positive numbers are not, using the Archimedian property of R. However, this does not work for the number zero so I am puzzled if this was an element of the intersection. Intuitvely I would say no but I cannot prove it. However, If I said: 1 > 0 what is obviously true I could divide by any positive n to get: 1/n > 0 and any negative -n to get -1/n < 0. Consequently 0 would be in this intersection but I'm not sure at all... Thank you for any help!
Best, Rafael
Let x be any positive number. By the Archimedean property, there exist an integer, n, such that n> x and thus 1/n< 1/x. x is NOT in the set (-1/n, 1/n) for that n and so is not in the intersection of all such sets.
Let x be any negative number. Then -x> 0 and as before there exist an integer, n, such that n> -x and thus, -1/n> x. x is NOT in the set (-1/n, 1/n) for that n and so is not in the intersection of all such sets.
The third case, x= 0, is the easiest. For any positive integer, n, n> 0 by definition of "positive" and so -n< 0. It follows that 1/n> 0 and -1/n< 0. That is -1/n< 0< 1/n for all n. 0 is in every set of the form (-1/n, 1/n) and so is in their intersection. The intersection of the collection of all open sets of the form (-1/n, 1/n) is just {0} which is a closed set and is not open. That is a reason why, while a union of any collection of open sets is open, on intersections of finite collections of open sets is open.