I don't understand your question but I understand this proof. It said

let a is g.l.b in S
(a is in S) ==> B (a) subset S
==> (a-e,a+e) subset S
==> a-e is in S
==> a isn't lower bound
(because a-e < a)
==> a isn't g.l.b
by contradiction we get a isn't in S
then a isn't g.l.b in S

I hope you understand more