Substitution in Big-O Estimate/General Result

Theorem
Let $X$ and $Y$ be topological spaces.

Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$.

Let $x_0 \in X$ and $y_0 \in Y$.

Let $f: X \to Y$ be a function with $\map f {x_0} = y_0$ that is continuous at $x_0$.

Let $g, h: Y \to V$ be functions.

Suppose $\map g y = \map O {\map h y}$ as $y \to y_0$, where $O$ denotes big-O notation.

Then $\map {\paren {g \circ f} } x = \map O {\map {\paren {h \circ f} } x}$ as $x \to x_0$.

Proof
Because $g = \map O h$, there exists a neighborhood $V$ of $y_0$ and a real number $c$ such that:
 * $\norm {\map g x} \le c \cdot \norm {\map h x}$ for all $y \in V$.

By definition of continuity, there exists a neighborhood $U$ of $x_0$ with $\map f U \subset V$.

For $x \in U$, we have:
 * $\norm {\map g {\map f x} } \le c \cdot \norm {\map h {\map f x} }$

Thus $g \circ f = \map O {h \circ f}$ as $x \to x_0$.