I need to prove the following:
1.) The intersection (infinity on top, n=1 on bottom) of the open interval (0,1/n) = the null set
2.) the intersection (infinity on top, m=1 on bottom) of the interval [m, infinity) = the null set
How do you show these? Why don't they adhere to the Nested Interval Property?
If you had the closed intervals,
Originally Posted by MKLyon
Then by the nested interval theorem, it would be a single point since lim 1/n---> 0
That point would be zero.
Since, these are open intervals that point (zero) is not included. Thus, the intersection is empty.
Here again is another way.
In a previous reply I gave you this lemma.
If e>0 then is a positive integer K such that (1/K)<e.
Looking at the intersection of the collection of sets (0,1/n), if it were not empty then it would contain some e>0. But e is not in set (0,1/K) in the collection, hence a contradiction. The intersection must be empty.
How would you go about proving the second one:
the intersection (infinity on top, m=1 on bottom) of the interval [m, infinity) = the null set.
Thanks for any help.
Well, once again if there were a number in the intersection of the [m,oo) that number would be an upper bound for the positive integers. That is a contradiction.