Equivalence of Definitions of Asymptotically Equal Sequences

Theorem
Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences in $\R$.

2 iff 3
Let $a_n - b_n = o(b_n)$.

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

Then

So $|a_n-b_n| \leq \dfrac{\epsilon\cdot |a_n|}{1-\epsilon}\leq 2\epsilon \cdot |a_n|$.

Thus $a_n-b_n = o(a_n)$.

The other implication follows by symmetry.