There are many places in programming where you don't care to prove properties of your program to this level of rigor; that's fine -- sufficiency is an important distinction: if a handful of examples are enough to convince you that your implementation is correct with regards to your specifications, then it's good enough. Does the file get copied to the right place? Cool.
However there are many more places where unit tests aren't sufficient. You can't express properties like, "this program can share memory and never allows information to escape to other threads." Or, with <= 10 writers all transactions will always complete. A unit test can demonstrate an example of one such case at a time... but you will never prove things one example at a time.