$X_i \sim N(\theta,1), \theta \in \Bbb Z$: $T=\left\lfloor \bar X_n \right \rfloor$ not consistent for $\theta$

Question

Let $\{X_i\}$ be a sample from a normal $N(\theta,1)$, where $\theta \in \Bbb Z$. Show the estimator $$ T=\left\lfloor \bar X _ n \right \rfloor, $$ is not consistent for $\theta$, with $\bar X_n$ the sample mean of an $n$-sample.

Let $\epsilon >0$, I should show $$ \lim_{n \to \infty}P\left( |T-\theta| > \epsilon \right) = \lim_{n \to \infty} P\left( \left|\left\lfloor \bar X _ n \right \rfloor - \theta \right| > \epsilon \right) \ne 0, $$ but the integer part function confuses me.
I don't see the distribution of the random variable in the middle term.

Answer 1

Hint: If $\bar X_n<\theta$, then $\left\lfloor \bar X_n \right \rfloor\leq \theta-1$. What is $P(\bar X_n<\theta)$?