Proof involving the Archimedean Property

In summary, if x is a real number, there is an integer m such that: -There is an integer m such that m≤x<m+1-If m=x, then m≤x. -By the Archimedean property, m+1>x and m+1>m, so m≤x<m+1.
  • #1
pissedoffdude
2
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
 
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  • #2
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?
 
  • #3
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
 

FAQ: Proof involving the Archimedean Property

1. What is the Archimedean Property?

The Archimedean Property is a mathematical principle that states that for any two positive real numbers, there exists a positive integer that is greater than the first number and less than the second number.

2. How is the Archimedean Property used in proofs?

The Archimedean Property is often used in proofs to show that a certain quantity or value is unbounded or tends to infinity. It can also be used to establish the existence of a limit or to prove inequalities.

3. Can the Archimedean Property be applied to other types of numbers?

Yes, the Archimedean Property can be extended to other types of numbers, such as rational and complex numbers. However, it is most commonly used with real numbers.

4. What are some real-life applications of the Archimedean Property?

The Archimedean Property has various real-life applications, such as in physics to calculate the speed and acceleration of objects, in economics to model supply and demand, and in engineering to determine the stability and convergence of numerical methods.

5. Is the Archimedean Property a fundamental concept in mathematics?

Yes, the Archimedean Property is considered a fundamental concept in mathematics as it is used in various fields and is an essential part of the real number system. It also has implications in other branches of mathematics, such as calculus and analysis.

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