circuit analysis - Why is the voltage of a capacitor equal to the voltage of a battery connected it?

I am confused by this question. If we connect a resistor in series to capacitor then the voltage will drop across the resistor and now voltage across the capacitor is less than the source, then why would capacitor charge till it has volts equal to the source?

and why in the below image when the capacitor is fully charged, its voltage is not same as source?