Let $X_1,...X_n$ random sample from $f(x;\theta)=I_{[\theta-\frac{1}{2};\theta+\frac{1}{2}]}(x)$. i) Show that $(X_{(1)},X_{(n)})$ is a confidence interval for $\theta$.ii) find your confidence level.
The first item I do not know how to show that a range is confidence interval.
In the second item I tried $$P(X_{(1)}\leq \theta \leq X_{(n)})=P(X_{(n)}\geq\theta)-P(X_{(1)}\leq \theta)$$.
But the solutions say that $P(X_{(1)}\leq \theta \leq X_{(n)})=P(X_{(n)}\geq\theta)-P(X_{(1)}\geq \theta)$
But I don't understand this inequality,where $X_{(n)}$ is the max and $X_{(1)}$ is the minimum