That is very strangely worded! Grammatically, you are saying "let the empty set be bounded". I think you meant to say "let S be a non-empty set such that [b]S[b] is bounded". Also, you haven't said what "E" is. I think you mean that,for all E> 0, there exist s1 such that s1< beta+ E.

Try "proof by contradiction". Since beta is a lower bound for S, there is no member of S less than beta. If there were also no s1 with s1< beta+ E, then no member of the set is less than beta+ E. That means that beta+E is also a lower bound for S and gives a contradiction.