00:01
For this problem, we are asked first, what is the least upper bound property for non -empty subsets of real numbers? so there are two parts for that first question.
00:11
We have that there's some non -empty subset.
00:15
Let's say s is a subset of r.
00:20
We have number x is an upper bound.
00:26
It's an upper bound if x is greater than or equal to s.
00:35
For all or that should be little s i'll make a change to that s or x is greater than or equal to little s for all little s in our set big s and we have that x is the least upper bound or or all abbreviated as l u b for s big s if x is an upper bound is an upper bound and x is less than or equal to y for all upper bounds y of s then does the least upper bound property hold for subsets of the rational numbers i'll put a little segmenting line here the answer here is no the reason why or we can illustrate the reason why let's say we have x is a rational number such that x squared is less than or equal to two.
01:50
That would be equal to the intersection of the rational numbers with the interval from negative root 2 up to root 2.
02:02
The reason why this doesn't work is that root 2 is not an element of the rational numbers and we have that the irrational numbers, irrationals are dense.
02:21
So you can always find an irrational number in between any pair of rational numbers.
02:28
So the illustration of this would be, let's say we have our set here, we have one over one, and we could say we have seven over five, and so on.
02:42
We have our dividing line here is root two.
02:46
And the thing is that for any possible rational number that we take, 7 over 5, for instance, we would be able to take another rational number, which is closer to the square root of 2...