$h \rightarrow 0$時，$o(h) = o(|h|)$

==定理==

$h \rightarrow 0$時，$o(h) = o(|h|)$

==證明==

若$f(h) = o(h)$，則$\lim \limits_{h \rightarrow 0} \frac{f(h)}{h} = 0$。由極限的$\varepsilon-\delta$定義，$\forall \varepsilon > 0, \exists \delta > 0, \forall h \in (-\delta, \delta) \backslash \{0\}, \left| \frac{f(h)}{h} - 0 \right| < \varepsilon$。於是我們有
$$\begin{align} \varepsilon &> \left| \frac{f(h)}{h} - 0 \right| \notag \\ &= \left| \frac{f(h)}{h} \right| \notag \\ &= \frac{|f(h)|}{|h|} \notag \\ &= \frac{|f(h)|}{||h||} \notag \\ &= \left| \frac{f(h)}{|h|} \right| \notag \\ &= \left| \frac{f(h)}{|h|} - 0 \right|. \notag \end{align}$$
這意味著
$$\lim_{h \rightarrow 0} \frac{f(h)}{|h|} = 0.$$
從而$f(h) = o(|h|)$。

若$f(h) = o(|h|)$，則$\lim \limits_{h \rightarrow 0} \frac{f(h)}{|h|} = 0$。由極限的$\varepsilon-\delta$定義，$\forall \varepsilon > 0, \exists \delta > 0, \forall h \in (-\delta, \delta) \backslash \{0\}, \left| \frac{f(h)}{|h|} - 0 \right| < \varepsilon$。於是我們有
$$\begin{align} \varepsilon &> \left| \frac{f(h)}{|h|} - 0 \right| \notag \\ &= \left| \frac{f(h)}{|h|} \right| \notag \\ &= \frac{|f(h)|}{||h||} \notag \\ &= \frac{|f(h)|}{|h|} \notag \\ &= \left| \frac{f(h)}{h} \right| \notag \\ &= \left| \frac{f(h)}{h} - 0 \right|. \notag \end{align}$$
這意味著
$$\lim_{h \rightarrow 0} \frac{f(h)}{h} = 0.$$
從而$f(h) = o(h)$。

綜上所述可得$o(h) = o(|h|)$。

(證明終了)