Proof 1:
The capacitive reactance (otherwise known as resistance) is described by a formulae Xc = 1/(2*pi*f), where pi = 3.14 and fc is the frequency. From basic electrical principles it's clear that for DC the frequency is zero. Therefore the capacitor offers infinite resistance to DC and allows ac to pass freely because the ac has a fixed or defined frequency.

Proof 2:
A microscopic view of it would be like this : the capacitor plates are separated by a little distance and hence when a DC supply is increased across it the plate connected across the +ve terminal of the battery and that connected across the -ve terminal of the battery is charged to the potential of the battery and hence it starts blocking the charge entering it form a DC source. Whereas from an ac source, the voltage changes continuously in magnitude and direction and hence the capacitor plate does not get a fixed charge to block any of the voltage or current that is alternating in nature. hence capacitor blocks only DC and acts as a normal conducting element for ac supply.