1. ## sets

can someone explain to me what the below means?

consider the set S= (a,b). here b= sup S. let ε > 0, find x ∈ S such that b-ε ≤ a.

to solve this, there are two cases
case 1:
ε≥ b-a. then b-ε≤ a and thus we can take any x of the interval (a,b).

case 2:
ε ≤ b-a then b- ε ∈ (a,b). we can take for x to be the midpt of (b-ε, b). clearly, x ∈ (a,b) and b-ε < x.

can someone explain to me what does the cases talk about becos i have no idea why it works..

thanks!

2. Hello alexandrabel90
Originally Posted by alexandrabel90
can someone explain to me what the below means?

consider the set S= (a,b). here b= sup S. let ε > 0, find x ∈ S such that b-ε ≤ a. I think this should be b-ε ≤ x.

to solve this, there are two cases
case 1:
ε≥ b-a. then b-ε≤ a and thus we can take any x of the interval (a,b).

case 2:
ε ≤ b-a then b- ε ∈ (a,b). we can take for x to be the midpt of (b-ε, b). clearly, x ∈ (a,b) and b-ε < x.

can someone explain to me what does the cases talk about becos i have no idea why it works..

thanks!
Study the diagram I've attached. I hope this will explain what's going on here. In each case you need to explain how to find a number, $\displaystyle x$, for which $\displaystyle b - \epsilon \le x$.