Proof by Contradiction

What is a proof by contradiction?

A proof by contradiction is a method of mathematical proof where we

1. Start by assuming the opposite (negation) of what we want to prove is true
2. Show this assumption leads to a logical contradiction
3. Conclude that since our assumption led to a contradiction, the original statement must be true

The power of this method lies in the fact that if assuming a statement is false leads to a contradiction, then it must be true - It is just very elegant!

Let's prove triangle inequality using a proof by contradiction!

The Triangle Inequality states that in any triangle, the sum of any two sides must be greater than the length of the remaining side. This applies to all triangles, not just right triangles.

This inequality has a simple physical interpretation: if you want to get from point A to point B, going directly (one side) is always shorter than going through a third point (sum of two sides).