Need clarification on Abbott's Proof of the Algebraic Limit Theorem for Sequences

Question

Here is the excerpt of his proof that I do not understand

I don't understand why he concludes that $|b_n| > |b|/2$ from this.

Answer 1

Observe that by the triangle inequality $|b|-|b_n|\leq |b_n-b|<|b|/2$. Then rearranging for $|b_n|$ you obtain the result.

To see that the trick with the triangle inequality, note $|b|=|b_n+(b-b_n)|\leq |b_n|+|b-b_n|=|b_n|+|b_n-b|$. Therefore subtracting $|b_n|$ from both sides you obtain the result.

Answer 2

By reverse triangle inequality $$\left||b_n|-|b|\right|\leq |b_n-b|<\frac{|b|}{2}.$$ So $$|b|-\frac{|b|}{2}<|b_n|.$$