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 x_{0} vs x. 2. Also, how does the above imply the case involving the orange; what "else" is there? Thank you very much!
The inequality |x - x_{0}| < ε^{2} doesn't specify whether x is to the right of x_{0} or to the left of it. That's the reason for the two inequalities.