Let be $X:\Omega\to[0,1,2,3,\dots, \theta]$ a discrete random variable which obeys a uniform distribution. We don't know $\theta$ and estimate it by the maximum of the sample $(x_1,x_2,\dots,x_n)$, denoted by $T(x_1,\dots,x_n)$. Show that the estimator is biased.
My approach is quite simple:
\begin{align*} &\mathbb{E}(T(x_1,\dots,x_n))\\ &=\sum\limits_{k=0}^{\theta}P(T(x_1,\dots,x_n)=k)\cdot k<\sum\limits_{k=0}^{\theta}P(T(x_1,\dots,x_n)=k)\cdot \theta\\ &=\theta\cdot\sum\limits_{k=0}^{\theta}P(T(x_1,\dots,x_n)=k)=\theta, \end{align*} which shows $\mathbb{E}(T(x_1,\dots,x_n))<\theta$. On the last equal sign I have used the fact that all probabilities sum up to $1$. I got zero points for that solution and am wondering what is wrong with that approach?