Can anyone explain the second bullet point on the following answer?
I am trying to understand how this inequality is proven:
$$\delta(A,B)=|A|+|B|=|A|+|B-C+C|\leq |A|+|C|+|B-C|=\delta(A,C)+\delta(C,B)$$
How does one conclude that the inequality is correct and also how are these absolute values manipulated? If anyone can explain this step by step please.
First of all a useful zero is filled in and then a simple application of the triangle inequality follows which is $$|x + y| \le |x| +|y|$$
In more simple steps: $$\begin{align} |B| &= |B + 0| \\ &= |B + (C-C)| \\ &= |(B-C) + C| \\ &\le |B-C| + |C|\end{align}$$
So it follows: $$|A| + |B| \le |A| + |B-C| + |C|$$