I know that vacuum cannot pump more than 10.3 meters in earths atmosphere with a non-submerged pump. After that the pressure equalizes with the weight of the water and it cannot suck it any higher.

However, what would happen if I had a long pipe that had check valves in it every 10 meters or so? I create the vacuum at the top, suck the water up 10 meters, then release the vacuum. The water tries to fall but it cant because the check valves. Then when I create the vacuum again, im now starting with a water level 10 meters higher than the first time. By staging check valves throughout the pipe it seems like I could pump as high as I wanted without a submerged pump.

Would this work? Why or why not?