Could someone please check my work

Classify each of the following sets as open, closed, neither, or both. 1. Let S minEN . Sis : Open Closed Neither 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 . Also, n E S and - EN ( -: E ) for any n E N. v ( mic ) nS * Dand N (mic ) n ( RIS ) * 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 S' U {0} by definition 3.4.3. Since bdS = SU {0} , then by definition 3.4.6, since S contains some of its boundary, namely S but not all of its boundary, namely {0} , then S is neither open or closed