Given any real numbers a and b such that a<b, prove that for any natural number n

[snes_nerd]

Given any real numbers a and b such that a < b, prove that for any natural number n, there are real numbers x1, x2, x3, ... , xn such that a < x1 < x2 < x3 < ... < xn < b.

The hint I was given says : Define xi recursively by x1 = (a+b)/2 and x(i+1) = (xi +b)/2. Prove that xi < xi + 1 < b, and use this result to prove by induction that a < x1 < x2 < x3 < ... < xn < b for any n in the natural numbers.

Okay so the hint doesn't really help me at all but I think I have an idea of what the question is asking. Let's suppose a = 1 and b= 10. Then there are natural numbers between these such that 1 < ... < 10. So going back to the original proposition I could say that there are real numbers x1, x2, x3, ..., xn such that a < x1 < x2 < x3 < .. < xn < x(n+1) < b. Not really sure where to go from their.

 

[Office_Shredder]

Okay so the hint doesn't really help me at all

The hint is a fairly explicit statement of how to solve the problem to be honest. It's worth going over it and identifying what is confusing you about it

 

[snes_nerd]

I am sure it is very straightforward once I actually know what it means. In fact, its probably really easy. I feel kind of dumb not seeing it, and I am sure once I see it, I will feel even more dumb.

Maybe its because I don't understand why x1 = (a+b)/2 and (xi+b)/2. (a+b)/2 tells me a number between a and b, and (xi+b)/2 tells me another number between a and b that's greater then x1. Cant really connect this to the proof

 

[Office_Shredder]

So if x₁=(a+b)/2 and x₂=(x₁_b)/2 we have:

a<x₁<x₂<b

and if x₃=(x₂+b)/2 what do we have?

 

[SammyS]

Given any real numbers a and b such that a < b, prove that for any natural number n, there are real numbers x1, x2, x3, ... , xn such that a < x1 < x2 < x3 < ... < xn < b.

The hint I was given says : Define xi recursively by x1 = (a+b)/2 and x(i+1) = (xi +b)/2. Prove that xi < xi + 1 < b, and use this result to prove by induction that a < x1 < x2 < x3 < ... < xn < b for any n in the natural numbers.

Okay so the hint doesn't really help me at all but I think I have an idea of what the question is asking. Let's suppose a = 1 and b= 10. Then there are natural numbers between these such that 1 < ... < 10. So going back to the original proposition I could say that there are real numbers x1, x2, x3, ..., xn such that a < x1 < x2 < x3 < .. < xn < x(n+1) < b. Not really sure where to go from there.
...

For your example in which a=0 and b= 10: What are the values you get for x₁, x₂, x₃, x₄, x₅, etc. ?

 

[snes_nerd]

So x3=(x2+b)/2 implies that a < x1 < x2 < x3 < b?.

 

[SammyS]

So x3=(x2+b)/2 implies that a < x1 < x2 < x3 < b?.

Be more specific.

Using the hint, and a=0, b=10:

x₁ = (a+b)/2 = 5

x₂ = (x₁+b)/2 = 7.5

x₃ = (x₂+b)/2 = 8.75

...​

 

[HallsofIvy]

If a< b then a+ b< 2b so (a+ b)/2< b.

If a< b then 2a< a+ b so a< (a+ b)/2.

a< (a+ b)/2< b.

Yes, that's the whole point of an "average"- it lies between the numbers.

 

[snes_nerd]

Proof

If a < b, then a + b < 2b (axiom or reasoning I can assume this?)

Then with simple algebra, (a+b)/2 < b (definition of multiplicative inverse)

If a < b, then 2a < a + b (axiom or reasoning I can assume this?)

Then a < (a+b)/2 ( definition of multiplicative inverse

Thus, a < (a+b)/2 < b

Okay that makes sense but this problem has to be solved with induction. And Sammy 0 and 10 is not in the actual problem. I was just throwing an example out. I still have to prove with induction that xi < xi+1 < b. It does make sense this way though although I HAVE to show it with induction for any n in the natural numbers.

 

[SammyS]

...

Okay that makes sense but this problem has to be solved with induction. And Sammy 0 and 10 is not in the actual problem. I was just throwing an example out. I still have to prove with induction that xi < xi+1 < b. It does make sense this way though although I HAVE to show it with induction for any n in the natural numbers.

And I was trying to point out how to use your example to help you understand how the hint that was given could be useful.