I am studying this proof for the uncountability of the reals using Cauchy sequences. In a later part of the proof, the author presents a equation that I'm not able to follow. To understand what the terms in the equation mean, I ask that you read the proof. The equation that I'm having trouble with is:
$$| b_n - a^k_n | = | (b_n - b_k) + (b_k - a^k_{N_k}) + (a^k_{N_k} - a^k_n) | \tag{1}\label{eq1}$$
$$| b_n - a^k_n | \geq | b_k - a^k_{N_k} | - | b_n - b_k | - | a^k_{N_k} - a^k_n |\tag{2}\label{eq2}$$
I do not follow how \eqref{eq2} follows from \eqref{eq1}. Is it using some property of modulus that I am not aware of? Or does it come from some part earlier in the proof? There is no relation that has been established between $b_n$ and $a^k_n$ earlier in the proof.
Could you please help me understand?