Prove that for any finite sequence $\{a_{k}\}_{k=1}^{n}$ of real numbers there is an index $m\in\{0,\cdots,n\}$ such that
\[ \left|\sum_{1\leq k\leq m}a_{k}-\sum_{m<k\leq n}a_{k}\right|\leq \max_{1\leq k\leq n}|a_{k}| \]

Proof:

设
\[ S_{m}=\sum_{1\leq k\leq m}a_{k}-\sum_{m<k\leq n}a_{k}\]
有
\[ S_{0}=-S_{n} \]
而数列$\{S_{m}\}$将会改变符号，于是可以选择一个$p$,使得$S_{p}$和$S_{p-1}$不同号，必然存在一个$p$使得
$|S_{p}|$或者$|S_{p-1}|$不超过$\displaystyle\max_{1\leq k\leq n}|a_{k}|$，要是不会这样，就是说对所有能让$S_{p}$和$S_{p-1}$ 不同号的那些$p$,都有
\[ |S_{p}|>\max_{1\leq k\leq n}|a_{k}|,|S_{p-1}|>\max_{1\leq k\leq n}|a_{k}| \]
那么
\[ 2\max_{1\leq k\leq n}|a_{k}|<|S_{p}|+|S_{p-1}|=|S_{p}-S_{p-1}|=2|a_{p}|\leq 2\max_{1\leq k\leq n}|a_{k}|\]
便得到了矛盾，所以结论成立。