(Tricky) Absolute Value Inequalities

vertciel

Hello everyone,

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

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



By definition, [tex] |x - x_0| < ε^2 \Longleftrightarrow -ε^2 < x - x_0 < ε^2 \Longleftrightarrow -ε^2 + x_0 < x < ε^2 + x_0 [/tex]

1. How does this imply the inequality in red?

[tex] \text{ Since } ε > 0 \text{ then } x_0 - ε^2 < x_0 [/tex]

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!

Mark44

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.

vertciel

Thank you for your response, Mark44.

Could you please explain the red box?

SammyS

It looks like that's exactly what he did !