[Math] A finite set has no limit points

Question

I am looking for an easy to remember proof to the statement "A finite set has no limit points". The best proof I found say that

Any finite set is composed of isolated points only. Since for any
isolated point there exists a neighborhood that does not contain any
other element of the set, a finite set cannot have any limit points.

Is this proof correct and is there a way to represent this mathematically?

Answer 1

I'm not convinced.

Definition: s is a limit point of S if every open set that contains s contains a second distinct element of S.

Consider a finite non-empty set X of two or more elements with the indiscrete topology. Every point is a limit point. (there are only two open sets in the indiscrete topology, {}, and X itself).