Analysis 1 (Terence Tao): Cauchy Sequence Lemma involving Triangle Inequality

Question

I put an orange arrow next to where I'm having trouble comprehending. I understand everything except the last sentence of the paragraph in question.

How is it that we can conclude from the triangle inequality that $\left\lvert b_n\right\rvert$ $\geq$ $\frac\epsilon 2$ for all n $\geq$ N?

Thanks all!

Answer 1

Hint: Rearrange the terms of triangle inequality.

$$|b_{n0}| \le |b_{n_0} - b_n| + |b_n|$$

Thus

$$|b_n| \ge |b_{n0}| - |b_{n_0} - b_n|$$