Prove that the long exact sequence in the statement of the snake lemma is exact at the term . The long exact sequence is: . The diagram is below.
what i don't like about homological algebra is "diagram chasing" ... it's really annoying! anyway, since the proof is quite long and it can be found almost anywhere, i won't bring it in here.
there's a very detailed and extremely easy-to-understand proof of the lemma in here. (page 3, from line 6 to the end of the page is the solution to your problem)