i need to prove that: 1) every infinite set and bounded has at least one accumulation point. basically i proved it by using the method of proving the theorem of bolzano for infinite sequences, basically if A is an infinite set and bounded, then for every x in A there's a,b such that a<=x<=b we can divide [a,b] into two parts [a,(a+b)/2] [(a+b)/2,b] at least one of them has an accumulation point, and that way we continue the same way as in bolzano theorem. 2) if E is a bounded set, then the set of accumulation points, E' is also bounded. basically what i did is as follows: let E be bounded, then for every[tex]x \in E[/tex] there exist a,b reals such that a<=x<=b and because E' is accumulation points set, for every x' in E' there exists x0 in E such that x'!=x0 and for every e, x0 in (x'-e,x'+e). if x'>x0, then x'-e<x0<x'+e x'<e+x0, a<=x0<x'<e+x0. if x'<x0, then x0-e<x'<x0<=b. either way, x' is bounded. are either of my proofs are correct? thanks in advance.