Prove that there is no unbiased estimator for $1/ \theta$ when we have one observation from $U(0,\theta)$.

Question

If we have a random sample $X_1,...,X_n$ from $U(0,\theta)$, then it can be proved that for $\frac{1}{\theta}$, an unbiased estimator is $\frac{n-1}{nX_{(n)}}$, where $X_{(n)}:=\max(X_1,...,X_n)$, which is also sufficient and complete and has density $n \theta^{-n}t^{n-1}$. Therefore, for $n=1$, there is no unbiased estimator for $\frac{1}{\theta}$. Is there an alternative proof for the non-existence of an unbiased estimator for $\frac{1}{\theta}$ when $n=1$?