Little-O Implies Big-O

From ProofWiki
Jump to navigation Jump to search

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.