(Tricky) Absolute Value Inequalities

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Homework Help Overview

The discussion revolves around proving that the function \(\sqrt{x}\) is uniformly continuous on the interval \([0, \infty]\). Participants are examining the implications of an absolute value inequality in the context of this proof.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand how the inequality \(|x - x_0| < \epsilon^2\) leads to specific implications regarding the values of \(x\) and \(x_0\). Questions are raised about the relationship between \(x\) and \(x_0\) and how this affects the proof.

Discussion Status

Some participants are seeking clarification on specific inequalities presented in the original post, while others are providing insights into the nature of the inequalities involved. The discussion appears to be focused on unpacking the reasoning behind the inequalities rather than reaching a consensus on the proof itself.

Contextual Notes

There is an emphasis on understanding the implications of the absolute value inequality, and participants are questioning the assumptions regarding the positions of \(x\) and \(x_0\) in relation to each other.

vertciel
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Hello everyone,

I'm posting here since I'm only having trouble with an intermediate step in proving that

\sqrt{x} \text{ is uniformly continuous on } [0, \infty].

1zfjwxs.png


By definition, |x - x_0| &lt; ε^2 \Longleftrightarrow -ε^2 &lt; x - x_0 &lt; ε^2 \Longleftrightarrow -ε^2 + x_0 &lt; x &lt; ε^2 + x_0

1. How does this imply the inequality in red?

\text{ Since } ε &gt; 0 \text{ then } x_0 - ε^2 &lt; x_0

However, I do not know more about x0 vs x.

2. Also, how does the above imply the case involving the orange; what "else" is there?

Thank you very much!
 
Last edited:
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The inequality |x - x0| < ε2 doesn't specify whether x is to the right of x0 or to the left of it. That's the reason for the two inequalities.
 
Thank you for your response, Mark44.

Could you please explain the red box?
 
vertciel said:
Thank you for your response, Mark44.

Could you please explain the red box?
It looks like that's exactly what he did !
 

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