Let $\{x_n\}$ be a sequence in $\mathbb{R}_+^\mathbb{Z}$. Define
\begin{align}
a_n&=\sup_{k \ge n} \{x_k\} \\
b_n&=a_n+\frac{1}{n} \\
c_1&=b_1 \text{ and } c_n=\max\left(c_{n-1}-\frac{1}{n},b_n\right)
\end{align}
Show
$$
\frac{1}{n} \ge c_n - c_{n+1} \ge \frac{1}{n+1}
$$
Writing \begin{align} c_n &\ge b_{n+1}, \\ c_n &\ge c_{n-1}-\frac{1}{n} \end{align} I get $$ c_n-c_{n+1} \le c_n - c_ n+ \frac{1}{n+1} = \frac{1}{n+1} \le \frac{1}{n} $$ I don't see how to get the lower bound.