General Topology – Balanced Word to Balanced (Sturmian?)

Let $ E $ to be a balanced finite word: for two subwords $ U, V $ the same length, the number of $ 1 $& # 39; sin $ U $ differs from the number of $ 1 $& # 39; sin $ V $ by at most one.

  • can $ E $ to be pursued to an infinite balanced sequence?
  • can $ E $ to be pursued until an infinite sequence of Sturmian?

This came when I tried to solve an exercise, where I have to show that the Sturmian sequences are dense in the space of balanced sequences (with respect to the usual topology of symbolic dynamics).