Prove that the interval -2, 4) is an open set. Solution Let x be an element of (-2,4). Then -2 < x < 4, so 4-x and x+2 are both positive. Let d = min(x+2, 4-x). Then the set (x - d\\2, x + d\\2) contains x, and is a subset of (-2,4) [since x - d\\2 >= x - 1/2(x+2) = (x-2)/2 > 2, and similarly x + d/2 <= x + 1/2(4-x) = (x+4)/2 < 4]. Hence (-2,4) is open..