Assuming that things are perfectly elastic at the subatomic level, an electron incapable of escaping the electric field of a proton will not come to rest at its surface. It will bounce. With no energy added to the electron, it will be stuck in limbo. It will neither combine with the proton to form a neutron nor escape into space.
The electron clouds that surround atomic nuclei are manifestations of this. Every cloud corresponds to a bouncing electron.
For atoms with more than two protons in their nuclei, there is not enough room for all of the electrons to bounce directly off the nucleus. Only two electrons can do this. Additional electrons bounce off of the electric field of the electrons closer to the nucleus. These electrons are attracted by the nucleus, but repelled by their fellow electrons.
The inner electrons are not free to bounce at any random frequency. They have to bounce at a frequency that resonates with the nucleus. Only specific harmonics are allowed.
The electrons farther out are in turn bound by the frequencies of the electrons closer to the nucleus. All the electrons are therefore bound directly or indirectly to the resonant frequencies allowed by the nucleus.
This is why electron clouds come in a limited number of allowed energy states.