Sometimes in proofs related to limits I see the use of something like this: $$|a-b| = \mathbf{|(a-c)+(c-b)|} \leq |a-c|+|c-b|.$$

What is the logic of doing such thing apart from the reason that it just works?

Edit: An example:
![enter image description here](https://i.stack.imgur.com/vo4oz.jpg)