Is it true that limit points of set of limit points don't exist? Is it true that limit points of set of limit points don't exist ?

I am not getting any example in which set of limit points has any limit point.
Perhaps the simplest example is the discrete topology, Let $X$={1,2,3,...,n}.We shall define discrete topology on it.Clearly each of $1,2,3,...,n$ are limits of {1},{2}, {3},..., {n} respectively.There collection  {1,2,...,n} has limit points $1,2,...,n$.
Perhaps the simplest example is the discrete topology, Let $X$={1,2,3,...,n}.We shall define discrete topology on it.Clearly each of $1,2,3,...,n$ are limits of {1},{2}, {3},..., {n} respectively.There collection  {1,2,...,n} has limit points $1,2,...,n$.