Use the lemma that says that if is a lin. dep. set, then there's some vector lin. dep. on the PRECEEDING ones (i.e., the 2nd one on the 1st one, or the 3rd on the 1st and 3nd one., etc.)
So: if is lin. dep. then there's some natural number r s.r. is lin. dep. on
As is a generating set for , it's clear from the above that cannot be lin. dep. for any (why?), so it must be that is lin. independent...and now end the argument.