How do you prove that the circumference of a circle is $2\pi r?$

Basically this is a definition thing.
$\pi$ is defined to be the ratio of the circumference of a circle over its diameter (or 2 times its radius).
This ratio is a constant since all circles are geometrically similar and linear proportions between any similar geometric figures are constant.
If you were looking for how the value of the ratio $\pi$ is calculated or how we know that the same ratio applies to the area of a circle, then those are different questions. (Ask if either of these are what you were looking for).