I noticed that quite often proofs go to extremities to show later that there is a contradiction. By extremities I mean that they introduce an object with properties that are either minimal or maximal. For example, when proving "given no vertices have odd degrees, the graph is eulerian", the proof starts with
Suppose $G$ is non-eulerian graph with at least one edge and no vertices of odd degree. Choose such a graph G with as few edges as possible. Let $C$ be a closed trail of maximum possible length in G.
Later we prove that there is a longer closed trail so there is a contradiction.
The point is I can't completely see when it is necessary to introduce those extremities. Because how I would start is just by assuming the implication is false, so there is a non-eulerian graph G with no vertices having odd degree, and get stuck there hence not reaching any contradiction. So, could you please explain what is the gut feeling or trigger that should hint about introducing those extremities? How to know if I need maximal or minimal property of some object? Should I introduce extremities for every object (e.g. why as few edges as possible, can't see how it affects the proof)? Do we even need them? Please elaborate on this, I just want to get some context about this technique.