00:01
In this problem, we want to give an example of a countable bounded subset s of r.
00:07
So some subset of r, that's countable bounded.
00:16
And the supremum and inf are in r minus s.
00:21
So in other words, they're not in s.
00:26
So we wanted to be a sort of discrete set because it has to be countable.
00:35
So, and then we also want that some kind of like infinite, the infamums and supremums are, they're sort of limits.
00:44
So we want the limits not to be in the set.
00:48
So one good example that kind of comes up in these questions a lot is 1 over n, where n goes from 1 to infinity.
00:58
So if you look at a number line, we have 0 here, 1 here, and then the set, it has 1.
01:08
One half, one third, one fourth, one fifth, and it's just going to get infinitely and infinitely close to zero.
01:20
So this set, its infamum, is zero, but zero is not in the set, right? zero is not in this set of one over n.
01:36
And this set is countable because it's indexed by the natural numbers n because indexed by the natural numbers n.
01:53
But the problem is that the supremum of this set is one, which is in this set, and that's an issue.
02:01
So what we can do to remedy this is let's just make a kind of copy.
02:05
Let's just make a kind of copy of this.
02:12
And let's just do 2 minus 1 over n from n equals 1 to infinity...