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

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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.
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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

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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
 
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