Showing an alternating sum is positive

The following observation is proved in this answer:

Let $d > 0$ and $U_1, \cdots, U_{n-1}$ be i.i.d. random variables uniformly distributed on $[0, 1]$. Write

$$U_{(1)} \leq \cdots \leq U_{(n-1)}$$

for the rearrangement of $U_1, \cdots, U_{n-1}$ in increasing order together with the convention that $U_{(0)} = 0$ and $U_{(n)} = 1$. Then

$$ \mathbb{P}\left( \max_{1\leq i \leq n} [U_{(i)} - U_{(i-1)}] \leq d \right) = \sum_{k=0}^{n} (-1)^k \binom{n}{k} \max\{0, 1-dk\}^{n-1}. $$

So the positivity of the quantity in question corresponds to the above observation with $d = 2/n$ and $n \geq 2$, which can be easily shown to be positive by geometric argument. For instance, we may bound the probability in the left-hand side from below by

$$ \mathbb{P}\left( \left| U_1 - \frac{1}{n} \right| < \epsilon, \cdots, \left| U_{n-1} - \frac{n-1}{n} \right| < \epsilon \right) $$

for sufficiently small $\epsilon > 0$, such as $\epsilon = \frac{1}{2n}$.