Let $\{ a_1,...,a_n\}, \{b_1,...,b_n\}\subseteq \mathbb{R}$. Then is it always true that: \begin{equation*} \left\vert\max_{i\le n} a_i - \max_{i\le n}b_i \right\vert\le \max_{i\le n} \left\vert a_i -b_i\right\vert \end{equation*} I just haven't been able to come up with a counter example yet...and it feels intuitively true...but I haven't been able to prove it.
If it is not is there a sufficient condition that would make this true?
Thanks
Let $j, k \in \{1,2,\dotsc,n\}$ be values such that $\max_{1 \leq i \leq n} a_i = a_j$ and $\max_{1 \leq i \leq n} b_i = b_k$. (If there is more than one choice for $j$ or $k$, just pick one of the candidate values, it doesn't matter.) The left hand side of the inequality is $|a_j - b_k|$.
Suppose first that $a_j \geq b_k$. Since $b_k \geq b_j$, we have $0 \leq a_j - b_k \leq a_j - b_j$. Therefore $|a_j-b_k| \leq |a_j-b_j| \leq \max_{1\leq i \leq n} |a_i-b_i|$.
Second, if $a_j \leq b_k$, then $|a_j-b_k| = b_k-a_j$. We have similarly $a_j \geq a_k$, so $0 \leq b_k-a_j \leq b_k-a_k$, and $|a_j-b_k| = b_k-a_j \leq b_k-a_k = |a_k-b_k| \leq \max_{1 \leq i \leq n} |a_i-b_i|$. (Or, just say that in this case you swap $a$ and $b$ to reduce to the previous case.)
