设数列 $\{a_{n}\}$，$\{b_{n}\}$，$\{c_{n}\}$ 满足：$b_{n} = a_{n} - a_{n+2}$，$c_{n} = a_{n} + 2a_{n+1} + 3a_{n+2}$（$n = 1$，$2$，$3$，$\cdots$），证明：$\{a_{n}\}$ 为等差数列的充分必要条件是 $\{c_{n}\}$ 为等差数列且 $b_{n} \leqslant b_{n+1}$（$n = 1$，$2$，$3$，$\cdots$）.