In the context of Poisson processes, At the top of page 301 of Billingsley's Probability and Measure (3rd Ed) we obtain the equality $$P[N_t=n,N_{t+s_i}-N_t=m_i,1\leq i\leq u]=P[N_t=n]P[N_{s_i}=m_i,1\leq i\leq u]$$ The author then says that by induction on $k$, the following equality follows from the above one for $0=t_0<t_1<\cdots<t_k$: $$P[N_{t_i}-N_{t_{i-1}}=n_i,1\leq i\leq k]=\prod_{i=1}^kP[N_{t_i-t_{i-1}}=n_i]$$
I don't understand how this follows nor how it implies independence of increments.
For the induction part, use the statement (23.13) k times by rearranging terms. Then take t=0 and n=0, you would find out the formula you provided. Finally by observing the joint distribution is product of separate functions of variables ni and the fact we know the distribution of N(ti-ti-1) is Poisson, we get simultaneously Nti-Nti-1 has the same distribution as N(ti-ti-1) and independence(try to write out the formula of marginal distribution and compute the product, then use the fact that Xi is independent iff the joint distribution is the product of marginal distribution by recalling the grouping theorem(Pi-Lambda) for independence) .