Our professor gave us as an exercise (for the first part) to find a ring structure on the set of a finite sequence of elements in a field $K$ such that the set of these sequences be isomorphic to $K[x]$. This was easy. But the second part of the exercise said: "Expand this statement to the set of all sequences of elements in $K$".
Now my question is: Was he maybe messing with us? Because, as far as I got about thinking about this exercise, even $K[x_1]\cdots [x_n]$ can be bijectively mapped only to finite sequences, so considerung (infinite) sequences would get me nowhere.