I'm trying to understand how all of the potentials during an action potential are created. My question specifically is about the sodium potassium pumps, however I would also be grateful if someone could verify if I have grasped the concepts correctly...

From what I understand, the resting membrane potential is at around -60mV, close to the equilibrium potential for potassium (from the Nernst equation) because the permeability of potassium is much greater than that of sodium, even though the permeability is low. However the resting potential is not as negative as the equilibrium potential for potassium because of some permeability of sodium and the permeability of potassium is very low so the tiny permeability of sodium has some impact in preventing the potential reaching the equilibrium potential of potassium. When the threshold potential of -50mV is reached, the voltage gated Na+ ion channels open, and sodium ions enter the cell, causing depolarisation. The potential rises to about +40mV, which is close to the equilibrium potential for sodium because the permeability for sodium is much greater than for potassium, however it doesn't quite reach the equilibrium potential for sodium because of some potassium ions leaving the cell. When the +40mV threshold is reached, sodium channels close and potassium channels open. Because of the now much higher permeability of potassium ions compared with when the membrane is at its resting potential, the potential becomes much closer to the equilibrium potential for potassium, which I think is about -90mV, and therefore the cell hyperpolarises. Then the potassium channels close.

Now I'm unsure as to how the resting membrane potential is reached from the hyper polarised state, and I also do not see where the action of the sodium potassium pumps comes into this...

Thank you in advance for any help :)

I'm trying to understand how all of the potentials during an action potential are created. My question specifically is about the sodium potassium pumps, however I would also be grateful if someone could verify if I have grasped the concepts correctly.

From what I understand, the resting membrane potential is at around $-60\rm\,mV$, close to the equilibrium potential for potassium (from the Nernst equation) because the permeability of potassium is much greater than that of sodium, even though the permeability is low. 

However, the resting potential is not as negative as the equilibrium potential for potassium because of some permeability of sodium and the permeability of potassium is very low so the tiny permeability of sodium has some impact in preventing the potential reaching the equilibrium potential of potassium. 

When the threshold potential of $-50\rm\,mV$ is reached, the voltage gated $\ce{Na+}$ ion channels open, and sodium ions enter the cell, causing depolarisation. The potential rises to about $+40\rm\,mV$, which is close to the equilibrium potential for sodium (because the permeability for sodium is much greater than for potassium). However, it doesn't quite reach the equilibrium potential for sodium because of some potassium ions leaving the cell. 

When the $+40\rm\,mV$ threshold is reached, sodium channels close and potassium channels open. Because of the now much higher permeability of potassium ions compared with when the membrane is at its resting potential, the potential becomes much closer to the equilibrium potential for potassium, which I think is about $-90\rm\,mV$, and therefore the cell hyperpolarises. Then the potassium channels close.

Now I'm unsure as to how the resting membrane potential is reached from the hyper polarised state, and I also do not see where the action of the sodium potassium pumps comes into this.