Expected Spacing Between Consecutive Uniform Order Statistics

Let $X_1, \ldots, X_n$ be iid $\operatorname{Uniform}(0,1)$ and let $X_{(0)} = 0$, $X_{(n+1)} = 1$. Show that $E[X_{(k+1)} - X_{(k)}] = \frac{1}{n+1}$ for every $k = 0, 1, \ldots, n$, and compute this value for $n = 4$.