Question

construct a bounded set of real numbers with exactly three limit points

Answer 1

\[ z = \{z: z= (e^{i 2\pi/2})^n + 1/n, n \in \Bbb N \}\]

Answer 2

if you want the set of real numbers then take argument of the e^(i2nPi/3) ... you will get 3 limit points, since the any ball of radius > 0 will contain infinite number of element of the set.