Prove that for all real numbers x there is a y n which case x<y

In summary, the statement "for all real numbers x there is a y in which case x<y" means that for any given real number x, there exists a real number y such that x is less than y. This can also be read as "for every x, there exists a y greater than x." This statement can be proven using the mathematical technique of induction, which involves showing that the statement is true for a base case and then using logical reasoning to show it is true for subsequent cases. It can be generalized to all types of numbers and is important in mathematics and science as a fundamental concept and in equations and models to describe relationships between variables. It does not directly relate to the concept of infinity, but can be applied to infinite
  • #1
Andrax
117
0

Homework Statement


prove that
([itex]\forall[/itex]x[itex]\in[/itex]R) ([itex]\exists[/itex]y[itex]\in[/itex]R) : x < y


Homework Equations





The Attempt at a Solution


x < y
y= x+1
then x<x+1
which is correct but I'm kinda not sure of this answer...
 
Physics news on Phys.org
  • #2
y=x+1 satisfies x<y so what wrong with it?
 
  • #3
hedipaldi said:
y=x+1 satisfies x<y so what wrong with it?

just wanted to make sure someone actually confused me saying it dosen't
 

1. What does the statement "for all real numbers x there is a y n which case x

The statement means that for any given real number x, there exists a real number y such that x is less than y. This can also be read as "for every x, there exists a y greater than x."

2. How can this statement be proven?

This statement can be proven using the mathematical technique of induction. This involves showing that the statement is true for a base case (such as x=0) and then using logical reasoning to show that if the statement is true for x, then it is also true for x+1.

3. Can this statement be generalized to other types of numbers?

Yes, this statement can be generalized to all types of numbers, including integers, fractions, and irrational numbers. As long as the numbers follow the rules of order (such as x

4. How does this statement relate to the concept of infinity?

This statement does not directly relate to the concept of infinity. However, it can be applied to infinite sets of numbers, such as the set of all real numbers. In this case, it would mean that for any real number x, there is another real number y that is greater than x, even if x is infinitely large.

5. Why is this statement important in mathematics and science?

This statement is important because it is a fundamental concept in mathematics and science. It allows us to make logical deductions and prove mathematical theorems. It is also commonly used in mathematical and scientific equations and models to describe relationships between variables.

Similar threads

Can Natural Numbers m and n Satisfy x + 1/m < y - 1/n?
    • Calculus and Beyond Homework Help
Replies
22
Views
332
Prove if ##x<0## and ##y<z## then ##xy>xz## (Rudin)
    • Calculus and Beyond Homework Help
Replies
2
Views
592
Prove ##(a+b) + c = a + (b+c)## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
3
Views
515
Proving Completeness property of ##\mathbb{R}## using Dedekind cuts
    • Calculus and Beyond Homework Help
Replies
20
Views
2K
Prove ##a + b = b + a## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
3
Views
808
Find f(x,y) given partial derivative and initial condition
    • Calculus and Beyond Homework Help
Replies
6
Views
547
Obtain a nonzero solution of ##y''-4y'+x^2(y'-4y)=0## by inspection
    • Calculus and Beyond Homework Help
Replies
7
Views
678
Prove if x,y are reals that |xy| = |x||y|
    • Calculus and Beyond Homework Help
Replies
4
Views
9K
Does a Smallest Real Number Exist for a Given Real Number?
    • Calculus and Beyond Homework Help
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top