Could someone please check my work using the definition and theorem provided

Definition: Ler s be a subset of RR. A point X in R is an inferior point of S if there exists a neighborhood Nofx such That NSS. Theorem: A ser S is open iff S= ints. Equivalently, S is open iff every point in S is an inferior point of S. Ler S and I be subsers of IR. Prove: inT (s) is an open set using The definition and Theorem above. Show That Hint: Vx E inT (S) FE>0 such That N(x; E) E inT (S). Ler int ( s ) = ( 2, 6 ) Let XE inT (S ) Ler E = Min Za, 6-a] => N ( x ; E) E inT (S) by definition Since x is arbitrary, >> any x is an interior point of int (S ). By The Theorem , Since every x is an interior point of int (S ), Then int (S ) is open. Thus, inT (S ) is an open set