Substitution in Big-O Estimate

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

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

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

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

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

Suppose $g(y)=O(h(y))$ as $y\to y_0$, where $O$ denotes big-O notation.

Then $(g\circ f)(x)=O((h\circ f)(x))$ as $x\to x_0$.

Proof
Because $g=O(h)$, there exists a neighborhood $V$ of $y_0$ and a real number $c$ such that:
 * $\Vert g(x)\Vert\leq c\cdot\Vert h(x)\Vert$ for all $y\in V$.

By Definition of Continuity, there exists a neighborhood $U$ of $x_0$ with $f(U)\subset V$.

For $x\in U$, we have:
 * $\Vert g(f(x))\Vert \leq c\cdot\Vert h(f(x))\Vert$

Thus $g\circ f=O(h\circ f)$ as $x\to x_0$.