MHB Can a Constant be Chosen to Satisfy an Inequality for All Real Numbers?

  • Thread starter Thread starter solakis1
  • Start date Start date
  • Tags Tags
    Inequality
AI Thread Summary
The discussion revolves around the existence of a constant A that satisfies a specific inequality involving the floor function for all real numbers. Participants question the role of x_0, debating whether it is a fixed number or if the proof should hold for any value of x_0. The suggestion is made to set A as the reciprocal of the floor value of x_0, provided that [x_0] is not zero. Clarification is sought on whether the proof must accommodate various values of x_0 or if it can focus on a specific instance. The conversation emphasizes the need for a clear understanding of the parameters involved in the inequality.
solakis1
Messages
407
Reaction score
0
Prove or disprove the following:
There exists $A$ such that for all $a>0$ there exists $b>0$ such that for all $ x$:

$|x-\ x_0|<b$ i mplies. $|\frac{1}{[x]}-A|<a$ where [x] is the floor value of x

Gvf
 
Mathematics news on Phys.org
solakis said:
Prove or disprove the following:
There exists $A$ such that for all $a>0$ there exists $b>0$ such that for all $ x$:

$|x-\ x_0|<b$ i mplies. $|\frac{1}{[x]}-A|<a$ where [x] is the floor value of x
How does $x_0$ come into this?
 
put A= $\frac{1}{[x_0]}$ and $[x_0]$ is not 0
 
You have not explained the status of $x_0$. Is it a fixed number given in advance, or are we supposed to prove that the result holds for all $x_0$?
 
With $x_0$ we usualy define a constant that can take several different values but not integers for example 2.3, 4.1,12.9 e.t.c. e. t. c
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Replies
2
Views
1K
Replies
1
Views
2K
Replies
2
Views
1K
Replies
2
Views
2K
Replies
4
Views
2K
Replies
1
Views
1K
Replies
4
Views
1K
Back
Top