What is the result of this endgame? Can Black draw this position or does White win after all:

[Title "Black to move"]
[StartFlipped "0"]
[fen "8/P4R2/8/6kp/3K4/6P1/r7/8 b - - 0 1"]


I need variations that clearly illustrate your answer.
White has a very simple strategy to win this position - 


Prevent the g-pawn from being captured for free by the Black king (the Black rook cannot capture it because it is tied to the defense of the a-pawn promoting on a8. White wins if White can win the h-pawn)
Move the White king over to support the promotion of the a-pawn. 
At the right moment, occupy the b-file with the White rook and support the g-pawn from b3. Then White's king can have a safe square on the b-file to promote the a-pawn.

 [FEN "8/5R2/P7/6kp/3K4/6P1/r7/8 w - - 0 1"]

 1. a7 Kg4 (1... Kg6 2. Rc7 Ra5 3. Kc4 Kf6 4. Kb4 Ra1 5. Kc5 Ke6 6. Rh7 Rc1+ 7.
 Kb6 Rb1+ 8. Ka6 Ra1+ 9. Kb7 Rb1+ 10. Kc8 Rc1+ 11. Kb8 Rb1+ 12. Rb7 Ra1 13. a8=Q
 ) 2. Rg7+ Kh3 3. Kc3 Kg2 4. Kb4 Kh3 5. Kb5 Ra3 (5... Rxa7 6. Rxa7 Kxg3 7. Rg7+
 Kf3 (7... Kh3 8. Kc4 h4 9. Kd3 Kh2 10. Ke2 h3 11. Kf2 Kh1 12. Rg1+ Kh2 13. Rg3
 Kh1 14. Rxh3#) 8. Rh7 Kg4 9. Kc4 h4 10. Kd3 Kg3 11. Ke2 h3 12. Kf1 h2 13. Rh6
 Kg4 14. Rxh2)6. Kb6 Rb3+ 7. Ka6 Ra3+ 8. Kb7
 Rb3+ 9. Ka8 Rb5 (9... Rb2 10. Rg5 Rb1 11. Rxh5+ Kxg3 12. Rh8 Kf4 13. Rb8 Ra1
 14. Kb7 Rb1+ 15. Kc7 Rc1+ 16. Kd7 Rd1+ 17. Ke7 Re1+ 18. Kf7 Ra1 19. a8=Q) 10.
 Rb7 Ra5 11. Rb3! Kg4 12. Kb8 Ra2 13. a8=Q