Does anybody else have trouble with this? I have fairly short fingers. They aren't especially thick, but the only way that I can get all three strings to ring properly is to use the back of my fingernails to hold them down on the frets. I'm tearing the fingernails up and the damage seems risky since they're there to protect the skin underneath.

I'm not quitting by any means and I'm practicing every day, working on the chords I know and practicing the in combination for about an hour every day. I'm also a college student and there's not a lot more time than that to spare since I'm doing a ton of reading and I'm just not the type of person who can learn by doing a bunch of constant reading.