Equivalence of Definitions of Asymptotically Equal Sequences

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

2 iff 3
Let $a_n - b_n = \map o {b_n}$.

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 $a_n - b_n = \map o {a_n}$.

The other implication follows by symmetry.