Graded field over integers

Question

In a $\mathbb Z$-graded field $R$, prove that $R=R_0$. Use the fact that all units in a graded domain are homogenous or otherwise.

Try: Let $x_n$ be any nonzero element. Then it is homogeneous and unit. Then there is $y_m$ such that $ x_ny_m=1 $. As $ 1 \in R_0$, $m=-n$. From here how can I show $ n=0$ ?

Answer 1

Observe that $x_n+y_{-n}$ is non-zero element and hence is a unit in $R$. So $x_n+y_{-n} $ is homogenous, which is possible only when n=0.