Transitivity of Big-O Estimates

Theorem
Let $X$ be a topological space.

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

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

Let $x_0\in X$.

Suppose $f=O(g)$ and $g=O(h)$ as $x\to x_0$, where $O$ denotes big-O notation.

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

Proof
Because $f=O(g)$ and $g=O(h)$, there exist neighborhoods $U$ and $V$ of $x_0$ and real numbers $c,d\geq0$ such that:
 * $\Vert f(x)\Vert\leq c\cdot\Vert g(x)\Vert$ for all $x\in U$
 * $\Vert g(x)\Vert\leq d\cdot\Vert h(x)\Vert$ for all $x\in V$.

By Intersection of Neighborhoods in Topological Space is Neighborhood, $U\cap V$ is a neighborhood of $x_0$.

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

Thus $f=O(h)$ for $x\to x_0$.