How can we observe / prove the existence of things smaller than the smallest observable object to the electron microscope?

There is EVIDENCE of their existence, and you can directly measure such things as their mass, charge, and cross-sectional area with that EVIDENCE. It isn't just a matter of sitting around the bar making up random theories. These theories come from experimental observation, and they predict the results of experiments. Correctly. To the last decimal point.

Read up on the work of Rutherford for some excellent examples of how this is done without having to directly see the particles in question.

So, technically no, you can't observe them.

We can't observe these things we call particles or quanta. That's why quantum mechanics was invented, to concentrate on what we can observe...the observables...duh. So we really can't see what's going on with those quanta, we can only infer from what we can see, like wave patterns on a screen.

The down side for this approach is that we are left with uncertainties about what's really happening at the so-called process level. And that's why QM is typically written as probability equations and models.

There are all kinds of proof, but they are proof by inference, not direct observations of the processes in many if not most cases.