7 is actually not a member of this interval.

For the same reason as 5 wasn't an element of this interval.

But nevertheless, 7 is still the least upper bound.

There is no smaller upper bound of that interval than 7.

So it is a least upper bound. So least upper bounds are not the same as

maximums, and in this case, 0 is a member of that set.

That interval is defined remember as a set of reals, such as 0 less than or

equal to x less than or equal to 1. And in this case, the endpoints are 0 and

1 are elements of the interval. So 0 is in there, and it's clearly the

minimum elements of that interval. Well, the answer is the first one is

correct. This is what it means to say that the

rational line is dense. Between any two rationals, you can find a

third one. The second one is actually true, but it

doesn't express density. It doesn't express density, because

whether or not there is an irrational number between two rationals is, is sort

of irrelevant. it’s, it’s the question is about the

rational line being dense. And this is actually making a statements

about the real line. So it's true, but actually irrelevant to

the notion of density of the rational line.

So that one's not true. I mean, that one's true, but it's not the

answer to the question. And this one that's actually expressing

the notion of completeness. Now, if by a least upper bound, we mean

least upper bound in the rationals and the rationals on Q, then that would say

that the rational line is complete. Which is false, if however we interpreted

this to main, every set of rationals that is bounded above has at least upper bound

in the real numbers, then that would be an instance of the completeness of the

real line. And this is sure about the existence of

least upper bounds is what distinguishes the reals from the rationals and it's

what makes the reals a very powerful system.

For doing advanced mathematics and calculus in particular, and and

demonstrate the, the fact that the rationals is not complete is what

demonstrates the [INAUDIBLE] the impoverished niche of the rationals in

terms of mathematics and doing things like calculus.

Okay, how did you get on? What I want to do now is well actually,

what I really want to do is introduce the beginnings of the subject known as real

analysis. Now, this isn't real anaylsis as opposed

to fake analysis. Real here is essential short for real

numbers. Well, for the real number system if you

like. It's the analysis of the real numbers.

And I'm going to began with a theorem, the rational line is not complete.

Now, if you've done that assignment, assignment 10.1 that I've asked you to

do,you should be familiar with what that means.

But let me remind you in case you are decided to play, if plate was in and go

ahead without doing that assignment. Well, let me just remind you that

completeness means, if A subset of reals has an upper bound, then it has a least

upper bound in the set of reals. That is that was completeness as a

product of reals. But as I mentioned at the time, these

notions also apply to any set. So in terms of the rationals,

completeness would mean if A is a set of rationals having, an upper bound then it

has at least upper bound in the rationals.

What this theorem says is that this property does not hold for the rational

numbers. Remember, [INAUDIBLE] for real numbers

here. But if I replaced r by q and talked about

the rational numbers then this property would not hold.

It does however hold for the real numbers.

Now, there's the completeness property for the real line.

we won't be able to prove that but, I'll be able to indicate how it's possible to

construct the reals in order to make it possible to prove that.