Hi Dejan,
yes you have defined the path f: N ---> {0,1,2,3,4,5,6,7,8,9} with f(n)=3, or 0.333... or 1/3 or 10/30. But you did it with a _finite_ definition. In fact you cannot find out from the covered nodes of the decimal tree that I present you, whether I used the path of 1/3 or not, can you? I do not deny that 1/3 exists, but all its nodes at finite levels of the tree can be covered by paths which differ from 1/3, namely by all its FISs. 0.3000..., 0.33000..., ... You can say: 1/3 is the path that has no zeros whereas all FISs have zeros. In _every_ FIS I can find an index with digit 0 …