The idea of mathematical proof is not only to present a argument that is logical. A proof is designed convince people, persuade them that a statement is true. Many proofs are so complex and require so much math knowledge that they can only convince only other mathematicians, or even only a small subset of mathematicians deep into a topic. But there are some proofs that need no equations to present. The Triangle Inequality is such a proof. Here is the statement and a picture that helps persuade us the statement is true.
Given any triangle, the sum of the lengths of any two sides must be longer than the length of the third side.
We are taught that the shortest path from point A to point B is the straight line segment that connects them. If instead I were to travel from A to D to B, obviously I would be taking a detour and that trip would be longer, and that trip length is the length of AD plus the length of DB.
That's it. That's the proof. Q.E. frickin' D., as we say in the math biz.
Okay, that's not completely it. Mathematicians also include the possibility of picking three points on the plane that don't actually make a triangle, like points A, B and C shown here. Since C is on the shortest path from A to B, in this example the length of the line segments AC and CB is equal to the length of AB. To include this possibility, the statement of the inequality is expanded.
Given any three points, the sum of any two of the lengths of the line segments connecting the points must be greater than or equal to the length of the third line segment.
In higher mathematics, the major split is between algebra and analysis. Algebra gets even tougher than what you remember in school, becoming much more abstract. Analysis is the abstraction of geometry and calculus, dealing with the ground rules in special cases where ideas of distance and area are changed to deal with shapes that bend and twist in multiple dimensions, and some stuff even weirder. Last June, I wrote about distance on a sphere, where a "straight line" isn't possible because the sphere gets in the way.
During my graduate career, I found algebra easier than analysis. Analytical proofs can get very tricky with some of the steps seeming to be completely arbitrary. But no matter how odd the definition of shortest path gets in a mathematical system, it always must include a version of the triangle inequality, the idea that if you are traveling from point A to point B, there is no such thing as a magical shortcut if you visit some special other point C before reaching B. It might be the same distance if C is "between" A and B, but it can't be shorter.
I remember reading proofs and feeling like I was slogging through mud, but then would come a step that would say, "now we can just use the triangle inequality", and I would feel like I could take a breath. All the odd and difficult stuff that came before was being done so we could take advantage of this powerful idea, so simple that an easy picture and a few sentences can convince us that it is true.