Proposition: Every point of an open subset E in R^2 is a limit point.
Proof: Assume by contradiction that there is an x in E such that x is not a limit point. Since x is in the open set E, it is an interior point, and has a neighborhood N_R(x) contained in E. Let 0 < r <= R, and let N be the set of all neighborhoods of radius r. Define a function f: (0,R] -> N as a :-> N_a(x). The function f is clearly onto, so the proposition is equal to showing that f is bijective (which would yield the equivalent statement that x is a limit point <=> N has no minimal element).
