What does Proximal-Proximal solve?

A subtle mistake when minimizing the sum of two functions

Proximal-proximal solves the wrong problem.
It's an easy mistake to make,
and it's 𝘢𝘭𝘮𝘰𝘴𝘵 enough to solve 'em.

When minimizing the sum
of two functions f and g,
it's tempting to apply the proximal of one
followed by the proximal of the other.

Even more deceptive,
this iteration actually converges.
Its solution is related to what we want.

But, to use proximals of both f and g,
we ought use Douglas Rachford splitting
to minimize the sum.