Prove that the complement of a dense open subset of R is nowhere dense in R

Let b be a limit point of a set A ⊂ R. Prove that if  ∈ > 0 then (b −  ∈,b +  ∈) contains an infinite number of points that are in A.

Let A be a nonempty subset of R that is both bounded above andbelow and let B be a nonempty subset of A. Prove or disprove eachof the following assertions.(a) inf(A) ≤ inf(B) ≤ sup(B) ≤ sup(A).(b) If inf(A) = sup(A), then A has exactly one element.(c) If inf(A) = inf(B) and sup(A) = sup(B), then A = B.

(b) Prove that the complement of a dense open subset of R is nowhere dense in R..