Ordered field/Raising to power/Growth behavior/Fact/Proof/Exercise

Let ${}K$ be an ordered field and ${}n\in \mathbb {N} _{+}$. Show the following statements.

1. The mapping

   $K_{\geq 0}\longrightarrow K,x\longmapsto x^{n},$

   is strictly increasing.

2. The mapping

   $K_{\leq 0}\longrightarrow K,x\longmapsto x^{n},$

   is for ${}n$ odd strictly increasing.

3. The Mapping

   $K_{\leq 0}\longrightarrow K,x\longmapsto x^{n},$

   is for ${}n$ even strictly decreasing.

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