Let \(x \in \mathbf{R}\), \(-|x| \leq x \leq |x|\).


For the absolute value function definition and other properties see here.

Proof:

We have two cases:

  • if \(x \geq 0\), then \(|x| = x \geq 0\). So \(-|x|\) is negative and we have \(-|x| \leq 0 \leq x = |x|\)
  • if \(x < 0\), then \(|x| = -x\) or \(-|x| = x\). So \(x = -|x| < 0\). But we also know that \(|x| \geq 0\) by the previous lemma. So putting everything together we have, \(-|x| = x < 0 \leq |x| \)

This was a little hard as well but I need to remember that (x) is negative so (-x) is positive. \(\blacksquare\)

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