In math or statistics, proof by example does not work. One example of a phenomenon does not prove anything. For example, because 2 is prime doesn’t mean that all even numbers are prime. In fact, no even numbers other than 2 are prime.
But in other areas proof by example is the best way to demonstrate something. One example is writing letters of recommendation. It is way more convincing when I get one example of something a person has achieved:
Kyle created the first R package that can be used to analyze terabytes of sequencing data in under an hour.
Than something much more general but with no details:
Bryan is an excellent programmer with a mastery of six different programming languages.
In mathematics it makes sense why proof by example does not work. There is a concrete result and even one example violating that result means it isn’t true. On the other hand, if most of the time Kyle crushes his work, but every once in a while he has an off day and doesn’t get it done, I can live with that. That’s true of a lot of applied statistical methods too. If it works 99% of the time and 1% of the time fails but you can discover how it failed, that is still a pretty good statistical method…