I need help proving the following statements. I've been doing this homework for 6 hours and I am finally burnt out. Any help is greatly appreciated.
Let R denote the set of all reals.
1) Let S be a subset of R and let x be an element of R. Prove that one and only one of the following three conditions holds:
a) x is an element of the set of all interior points S (int S)
b) x is an element of int (R\S)
c) x is an element of the set of all boundary points S (bd S) = bd (R\S)
2) Let S be a bounded infinite set and let x = sup S. Prove: If x is NOT an element of S, then x is an element of S' (set of all accumulation points of S).
3) Prove: bd S = (cl S) intersection [cl (R\S)].
Thank you. Any help would be awesome.