Understanding the Proof of the Existence of Rational Numbers in Any Interval

[3ephemeralwnd]

Homework Statement

theorem: There is at least 1 rational # in any interval

Say a<b and let I = (a,b).

Let n be a positive integer so large that 1/n < b-a and consider the numbers k/n (where K is an integer). Then there will be a K such that k/n EI



I'm just having a bit of trouble understanding this proof, it just seems to lack a lot of.. explanation

I don't understand why n has to be a positive integer so large that 1/n<b-a .. and where does K come from?

 

[Mark44]

Maybe it will help your understanding if I use numbers rather than variables. Let's say that a = 3/4 and b = 7/8, so I = (3/4, 7/8)

7/8 - 3/4 = 1/8, so let's take n = 9. Then 1/9 < 7/8 - 3/4.

Now consider the multiples of 1/9.
1/9
2/9
3/9
4/9
5/9
6/9
7/9 <<<
8/9
9/9

Of these multiples of 1/9, 7/9 is in the interval I.

 

[3ephemeralwnd]

Thank you! i get it now..

would this also mean that there are an infinite number of IRRATIONALS on every interval "I" as well? ifso, how would i go about stating that theorem?

 

[Mark44]

That's true too. I think you could use the same idea: find an irrational number r that is smaller than the length of the interval, then multiply it by an integer n that is large enough so that a < nr < b.