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If b_n is an infinite sequence that converges to B such that B isn't 0, then there is a positive number M and a positive integer N such that if n>= N, then |b_n| >=M

I am trying to understand the proof but there is a line I don't understand enter image description here

I don't understand the circled part.

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    This looks like the reverse triangle inequality – 2017-02-22
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    I just can't see the reasoning behind it. I tried writing it down but it's not making senseZ – 2017-02-22
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    I mean they're using a theorem which you should look up – 2017-02-23
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    Namely if $x, y$ are numbers, then $|x - y| \geq | |x| - |y| |$ – 2017-02-23
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    @D_S do you mind providing me the steps just to see it being used? Like the additional in between steps? – 2017-02-23
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    Sure, take $x = b_n - B$ and $y = -B$ – 2017-02-23
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    $x = (x - y) + y$, so by the triangle inequaility $|x| \le |x - y| + |y|$ and therefore $|x| - |y| \le |x - y|$. Similarly $|y| - |x| \le |x - y|$. Therefore $\left| |x| - |y|\right| \le |x - y|$. – 2017-02-23

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