let the empty set not equal S which is a subset of R be bounded let beta= infS and E>0 then prove that there exsits s1 in S such that s1<beta+E
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.
if we assume that there does not exist an s1 in S such that beta +E > s1
s is greater than or equal to beta+E for all s in S
beta+E is a lower bound of S
beta is greater thab or equal to beta+E
0 is greater then or equal to E
now i am stuck... am i even going in the right direction with this