prove $S_u(A)⊆\bar A$

Question

Let A be a subset of $R$. Let us call $p$ an upper sticking point of $A$ if there is some subset $B ⊆ A$ such that $p = supB$. Let us write $S_u(A)$ for the set of all upper sticking points of $A$. it is always true that $A ⊆ S_u(A)$, since any $a ∈ A$ is the supremum of the one-pointsubset ${a}⊆ A$. Prove that $S_u(A)⊆ \bar A$, and give an example to show that the equality $S_u(A) =\bar A$ does not hold in general

Answer 1

If $p \in S_u(A)$ you know that you can approximate $p$ by points in $A$ so that $p \in \overline A$. To get a counterexample, consider the collection $\frac 1 n$ for $n>0$.