Show every bounded infinite set has a maximum limit point and a minimum limit point.

Show every bounded infinite set has a maximum limit point and a minimum limit point.

Here is my thought even if it is not formal

Let $S$ be bounded and infinite set.

Bolzano–Weierstrass theorem: Every bounded and infinite set has a limit point.Since it is bounded by completeness property(Can I apply?) the set has least upper bound(Sup(S)) and greatest lower bound(Inf(S)). Now my claim is that maximum limit point$=Sup(S)$ and minimum limit point$=Inf(S.)$I need someone to tell me how to proceed.