Let us assume the capacitor initially uncharged with switch open.
After the switch is closed, the battery begins to charge the plate of the capacitor and charge passes through a Resistor. As the capacitor is being charged the circuit carries a charging current the charging process continues untill the capacitor charged to its maximum equilibrium value, a = cv where v is the maximum voltage across the capacitor once the capacitor is fully charged, the current in the circuit is zero. The current approaches '0' at t = ∞ , because there is no voltage drop across the capacitor is equal to the Battery voltage. Hence at t = ∞, the capacitor act as an open circuit