Maybe it is not really suitable to undergrads (unless they are really problem solving-oriented), but there is a nice proof that $$\prod_{1\leq i \lt j\leq n} \frac{x_j-x_i}{j-i}$$ is integer for all integer sequences $(x_k)_{k=1}^n$ using Vandermonde determinants. The idea is reducing the thesis to the fact that the matrix $$\begin{bmatrix}1 & 1 & \dots \newline \binom{x_1}{1} & \binom{x_2}{1} & \dots \newline \binom{x_1}{2} & \binom{x_2}{2} & \dots \newline \vdots & \vdots & \ddots \end{bmatrix}$$ has integer entries, and thus integer determinant. After clearing the denominators (which give the factor $\prod \frac{1}{j-i}$), one can transform the resulting determinant to a Vandermonde with basic row operations.