Triangle Inequality and intersecting lines

Question

I'm looking at a proof and as part of the proof they say that line segment $f_iW_n$ has shorter length than $f_iW_m$, and that $f_kW_m$ has shorter length than $f_kW_n$. They said that this is true by the triangle inequality. Is there a property with intersecting lines that I'm missing?

The question is an extremal principal problem that asks us to form a bijective map between $2n$ points where $n$ points, indexed by $f$ are mapped to the other $n$ points, indexed by $W$m, such that no lines between these points intersect. There's the added condition that no $3$ lines are collinear. Here's the proof, but the only part relevant to my question is where they use the triangle inequality.

Answer 1

The solution in your image is correct, but your own interpretation is wrong.

It is not true in general that

$f_iW_n$ has shorter length than $f_iW_m$, and that $f_kW_m$ has shorter length than $f_kW_n$.

What is true in general is that the combined lengths of $f_iW_n$ and $f_kW_m$ is shorter than the combined lengths of $f_iW_m$ and $f_kW_n$, and this is indeed due to the triangle inequality:

In a triangle, the length of any side is less than or equal to the combined lengths of the other two sides.