Could someone please check my work

Find the boundary of the following sets: (Note that 0) is the empty set and {0} is the element 0.) 1. Let S = in E N . The boundary of S is: O Explain: Of S but O E N(0; E) . Since N(0; E) n S + 0 at some - E S where n is sufficiently large, and since N(0; E) n (R\\ S) # 0, then by definition 3.4.3 for the boundary point of S , {0 } is a boundary point of S . n Also, for any n E N , - E S and - EN(~: ). n Since N ( ;= ) ns / Dand N ( "; c ) n ( R \\S) + 0, then by definition 3.4.3 for the boundary of S , S is in the boundary of S . Therefore, the boundary of S is SU {0} by definition 3.4.3