Question about unique representation of reals The wiki on construction of real numbers from Cauchy sequences says 

A practical and concrete representative for an equivalence class representing a real number is provided by the representation [of Cauchy sequences] to base b. For example, the number π = $3.14159$... corresponds to the Cauchy sequence $(3,3.1,3.14,3.141,3.1415,...)$.

But you can just as easily write it as, for example, $(150,34,5, 4.1, 3.17, 3.15, 3.14,  3.1415, ...)$. That sequence is Cauchy, and it is equivalent to $(3,3.1,3.14,3.141,3.1415,...)$ because their difference tends to zero.
So how is $(3,3.1,3.14,3.141,3.1415,...)$ a unique representation (in base 10)?

In this construction reals are equivalence classes, as you understood. What your quote says is that one method to take a representative of a class is to take a base b development of the number you want to get. What you have is indeed a member of $\pi$ (seen as an equivalence class), Wikipedia just gave you some canonical representative.
