Please Login to access more options.

Problem 66: (Triangle Inequality)

For any real numbers $u$ and $v$, let $d(u,v)$ be the distance between $u$ and $v$, which means $d(u,v)=|u-v|=|v-u|$.

  1. Let $a,b,c\in\mathbb{R}$. Prove that $d(a,b)\leq d(a,c)+d(c,b)$.
  2. Let $x,y\in\mathbb{R}$. Use the previous result to prove that $|x+y|\leq |x|+|y|$.
Both facts above we call the triangle inequality. Both facts basically state that the distance from point $A$ to point $B$ is less than or equal to the distance traveled if you take the shortest route from $A$ to $B$ that must also pass through a third point $C$. Equality holds if $C$ is already on the shortest path from $A$ to $B$, otherwise the distance must increase.

The following pages link to this page.

Here are the old pages.