Show that if $(b_n) \to b$, then the sequence of absolute values $\left| b_n \right|$ converges to $\left| b \right|$.
By the triangle inequality, $\left| b_n \right| = \left| b_n - b +b \right| \leq \left| b_n - b \right| +\left| b \right|$.
Thus $\left| b_n \right| - \left| b \right| \leq \left| b_n -b \right|$,
and in fact $\left| \left|b_n \right| - \left| b\right| \right| \leq \left|b_n - b \right|$...
Why exactly is the "and in fact" line justified? That is, why are you allowed to just take the absolute value of $\left| b_n \right| - \left| b \right|$ and assume the inequality still holds?
As someone has already kindly pointed out to me in another post, this is the reverse triangle inequality. My understanding is that instead of applying that theorem here, the author sort of "proves" it as part of the solution. So I guess I am looking for an explanation of that step in the proof for the (in a limited sense) general case of the reverse triangle inequality... Please let me know if I need to be more clear or expand somewhere, otherwise, thank you for your help!