Firstly I think that

Why aren't there any other limit points? Between 1/n and 1/(n-1) there are infinitely many points of T and they are of the form 1/n + 1/m with m>n(n-1) and for any real number r between 1/n and 1/(n-1) there is a minimun distance to a point of T since r falls between two numbers of of form 1/n + 1/m