From 'Counterexamples in Analysis' by Bernard R. GelbaumJohn ,M. H. Olmsted i studied this:
The part underlined in red is unclear to me.If $c\in (A_n\setminus A_{n-1})\setminus I(A_n\setminus A_{n-1})$ ($I$ denote interior of a set) then $c\in\partial (A_n\setminus A_{n-1})$.
Then every neighbourhood of $c$ has points in $A_n\setminus A_{n-1}$ and in $\mathbb{R}\setminus(A_n\setminus A_{n-1})$.Then?