Simple inequality involving $\max(\cdot)$

Question

Let $x_1, \ldots, x_n$ be real numbers. Is it true that

\begin{equation} \max_{1 \leq i \leq n} |x_i| = \max \{ \max_{1 \leq i \leq n} x_i, \max_{1 \leq i \leq n} -x_i\} \ ? \end{equation}

My intuition is that the maximum in absolute value is either the smallest negative or the biggest positive number among $x_1, \ldots, x_n$, which motivates the given equation.

Any feedback is much appreciated.

Answer 1

\begin{align} \max_{1 \leq i \leq n} |x_i| &= \max_{1 \leq i \leq n} \left(\max\{x_i,-x_i\}\right)\\ &=\max\left\{ \{x_i,1 \leq i \leq n\}\bigcup\{-x_i,1 \leq i \leq n\}\right\}\\ &= \max \left\{ \max_{1 \leq i \leq n} x_i, \max_{1 \leq i \leq n} -x_i\right\}. \ \end{align}