Don't understand inequality in order to prove Algebraic Limit Theorem I'm self-studying from the book Understanding Analysis by Stephen Abbott and I'm stuck on Theorem 2.3.3 on page 45, i.e., the Algebraic Limit Theorem. 
In particular, letting $\lim a_n = a$ and $\lim b_n = b$, then I'm trying to follow the proof that $\lim (a_n/b_n) = a/b$ provided $b \neq 0$. The author writes that we choose an $N_1 \in \mathbb{N}$ such that $|b_n - b| < |b|/2$ for all $n \geq N_1$, which I understand. But then the author states that this implies that $|b_n| > |b|/2$, but I don't understand why this is implied.
So far, I've tried using the triangle inequality to write:
$$
|b_n - b| \leq |b_n| + |-b| = |b_n| + |b|
$$
but then I don't know how to continue, or maybe I'm not even on the right path. 
Any help is much appreciated.

Try using the triangle inequality as
$$|b| = |b - b_n + b_n| \leq |b - b_n| + |b_n|$$
Then use the assumption on $|b - b_n|$
