How does the proof connect the principles to show the elementary inequality?

[duke_nemmerle]

I'm looking over a proof and I'm wondering from which principles does it follow that
$$\mid a - b \mid < 1 \to \mid a \mid < \mid b \mid + 1$$

I can see that $$|a - b | \le |a| + |-b| = |a| + |b|$$ and that $$|a| - |b| < |a| + |b|$$ but I just can't connect the dots.

 

[duke_nemmerle]

I'm looking over a proof and I'm wondering from which principles does it follow that
$$\mid a - b \mid < 1 \to \mid a \mid < \mid b \mid + 1$$

I can see that $$|a - b | \le |a| + |-b| = |a| + |b|$$ and that $$|a| - |b| < |a| + |b|$$ but I just can't connect the dots.

Ahh, got it $$|a| = |(a+b)-b| \le |a-b| + |b|$$ which means $$|a| - |b| \le |a-b|$$ the result immediately follows