10. Let the fractions of the $n$-th order Farey sequence be $0=a_{1} / b_{1}<a_{2} / b_{2}<\cdots<a_{k} / b_{k}=1$. Prove:
(i) $k=1+\sum_{m=1}^{n} \varphi(m)$;
(ii) $\sum_{j=1}^{k} a_{j} / b_{j}=k / 2$;
(iii) $\sum_{j=1}^{k-1} 1 /\left(b_{j} b_{j+1}\right)=1$;
(iv) $\max _{1 \leqslant j<k}\left(a_{j+1} / b_{j+1}-a_{j} / b_{j}\right)=1 / n$,
$\min _{1 \leqslant j<k}\left(a_{j+1} / b_{j+1}-a_{j} / b_{j}\right)=1 / n(n-1)$.