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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.

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Try using the triangle inequality as

$$|b| = |b - b_n + b_n| \leq |b - b_n| + |b_n|$$

Then use the assumption on $|b - b_n|$

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