Little-O Implies Big-O
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Theorem
Sequences
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.
Let $a_n = \map \oo {b_n}$ where $\oo$ denotes little-$\oo$ notation.
Then $a_n = \map \OO {b_n}$ where $\OO$ denotes big-$\OO$ notation.
General Result
Let $X$ be a topological space.
Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$
Let $f, g: X \to V$ be mappings.
Let $x_0 \in X$.
Let $f = \map \oo g$ as $x \to x_0$, where $\oo$ denotes little-$\oo$ notation.
Then $f = \map \OO g$ as $x \to x_0$, where $\OO$ denotes big-$\OO$ notation.