let f be a function defined on E, a subset of R^n. lim (x -> x0) sup f(x) = M if and only if there exists a sequence {x_n} in E such that x_n -> x0 and f(x_k) -> M. how would you prove this fact? i thought that for a limit to exist it can approach the limit point from any direction, but this says that if you approach from one direction then the limit exists?