Index return identity over a basket change

Prove that if the divisor is adjusted by (1.8) at every basket change, the index return from $t_0$ to $t_n$ equals $\prod_{s=1}^{n} \big(1 + \sum_i w_{i,s-1} R_{i,s}\big) - 1$, where the weights at each date are those of the basket in force over the next day. Explain in words why an index fund that holds the index weights and trades only at basket changes, at closing prices, tracks the index exactly before costs.