What is the intuition behind this triangle inequality "trick"?

Question

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:

Answer 1

The idea is that you'd like to bound the distance between $a$ and $b$, but you know that they're both less than a certain distance away from some $c$. So, they can't be too far apart, since they're both 'close' to $c$.

Answer 2

Well, this is a proof that the "straight" distance between $P$ and $Q$ will always be less or equal to any sum of distances between any intermediary points between $P$ and $Q$.

In proofs about limits presumably you might know stuff about $|a-c|$ and $|c-b|$ but not between $a$ and $b$.

Without context this is a pretty vague question.

Answer 3

What is the intuition behind this triangle inequality “trick”?

For the context of limits, I would say

To ensure that the base of a triangle is small, it is enough to ensure that the other two sides are small.

For example, in the said context, we usually want to show that $|f(x)-L|$ is small (less than $\varepsilon$), provided that a certain condition is satisfied (usually, provided that $x$ runs over a small interval). The way we can do this is by showing that the "sides" $|f(x)-c|$ and $|c-L|$ are small (both at most $\varepsilon/2$), for some suitable $c$ (which usually depends on $x$ and whose relation with $f(x)$ and $L$ is known).