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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?

Randall
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Mandar M.
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    That is simply the (reverse) triangle inequality, in the form $|x+y+z| \ge |x| - |y| - |z|$. See for example https://math.stackexchange.com/q/2825288/42969. – Martin R Apr 21 '22 at 16:44
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    Clicking on the modular arithmetic tag leads to a page with this: "Modular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation a≡b(modn) which means that n divides a−b". The modulus here is not related to it so that tag should be removed. – Predrag3141 Apr 21 '22 at 16:53

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