Are All Periodic Signals Power Signals? Example of Non-Periodic Energy Signal Needed

1)All periodic signal are power signal but all power need not to be periodic signal . We have familiar example step signal.but i cant agree with that see my below derivation(attached file) and correct me

2)all non periodic signal are energy signals but vice versa is not true. could you give me an example for this .

Firstly a signal that posseses energy has the potential to do work - it can convert that energy into heat or just pass that energy on to something else. Energy isn't lost, it either remains "potential" energy or gets transferred to something else partially or wholly.
When energy is taken from an object or a component (such as an inductor or capacitor or spring or flywheel), the amount of energy taken divided by the time elapsed to take that energy is power. Over a long time period this is often called "average power"
Therefore, for a signal possessing energy to have zero power I guess you could say that it is not losing or gaining any energy i.e. it is in equilibrium.
By contrast, if something is giving off power and is still in energy equilibrium it must have infinite energy.

Firstly a signal that posseses energy has the potential to do work - it can convert that energy into heat or just pass that energy on to something else. Energy isn't lost, it either remains "potential" energy or gets transferred to something else partially or wholly.
When energy is taken from an object or a component (such as an inductor or capacitor or spring or flywheel), the amount of energy taken divided by the time elapsed to take that energy is power. Over a long time period this is often called "average power"
Therefore, for a signal possessing energy to have zero power I guess you could say that it is not losing or gaining any energy i.e. it is in equilibrium.By contrast, if something is giving off power and is still in energy equilibrium it must have infinite energy.
For example, a non-rechargeable battery is an energy signal. It holds a finite amount of energy. It can supply some power only for a limited time, but if you compute the average power over your lifetime you will get a very small number.
A power signal is always adding some energy, so integrated over all time (to get energy) will give an infinite result.