Equivalence of Definitions of Asymptotically Equal Sequences

Theorem
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $\R$.

Definition $(2)$ iff $(3)$
Let $a_n - b_n = \map \oo {b_n}$.

Let $0 < \epsilon < 1/2$.

Then:

So:
 * $\size {a_n - b_n} \le \dfrac {\epsilon \cdot \size {a_n} } {1 - \epsilon} \le 2 \epsilon \cdot \size {a_n}$

Thus:
 * $a_n - b_n = \map \oo {a_n}$

The other implication follows by symmetry.