Proof: Since there are $m$ congruence classes modulo $m$ given by (1), there must be two numbers among the $m+1$ numbers that belong to the same congruence class modulo $m$. These two numbers are congruent modulo $m$. This proves (i). By selecting a number $x_{r}$ as a representative in each congruence class $r \bmod m (0 \leqslant r < m)$, we obtain $m$ numbers $x_{0}, x_{1}, \cdots, x_{m-1}$ that are pairwise incongruent modulo $m$. This proves (ii).