Proof involving the Archimedean Property

  • Thread starter Thread starter pissedoffdude
  • Start date Start date
  • Tags Tags
    Proof Property
pissedoffdude
Messages
2
Reaction score
0

Homework Statement


If x is a real number, show that there is an integer m such that:
m≤x<m+1
Show that m is unique

Homework Equations


Archimedean Property: The set of natural numbers has no upper bound

The Attempt at a Solution


I'm having trouble with showing that m is unique. If x is a real number, I can find integers that are smaller and bigger than it. If m=x, then m≤x. By the Archimedean property, m+1>x and m+1>m, so m≤x<m+1
 
Physics news on Phys.org
Yes, that's fine if x happens to be an integer (that's what you imply when you say "if m= x". What if it isn't?

Whatever x is, you are correct when you say that the Archimedean property says that there exist integers larger than or equal to x. What can you say about the set of integers larger than or equal to x?
 
HallsofIvy said:
Yes, that's fine if x happens to be an integer (that's what you imply when you say "if m= x". What if it isn't?

Whatever x is, you are correct when you say that the Archimedean property says that there exist integers larger than or equal to x. What can you say about the set of integers larger than or equal to x?

Thanks for the advice.

So if I understand correctly, we have three cases:
1) x is an integer. Then, we can say m=x
2) x is rational. Then by the Archimedean property, we an find integers that are strictly greater and less than x, so we can let m be an integer such that m=x+1/2, then m<x<m+1 and x is halfway point here
3) x is irrational, then suppose we have the interval (m, m+1) where m is an integer, we let x be an irrational number somewhere in that interval
 
Thread 'Use greedy vertex coloring algorithm to prove the upper bound of χ'
Hi! I am struggling with the exercise I mentioned under "Homework statement". The exercise is about a specific "greedy vertex coloring algorithm". One definition (which matches what my book uses) can be found here: https://people.cs.uchicago.edu/~laci/HANDOUTS/greedycoloring.pdf Here is also a screenshot of the relevant parts of the linked PDF, i.e. the def. of the algorithm: Sadly I don't have much to show as far as a solution attempt goes, as I am stuck on how to proceed. I thought...
Back
Top