I have a discrete math page under construction at NYU. The videos are not public, but I have mirrored the most relevant ones here.

• Exaggerate and Simplify:
  http://research.engineering.nyu.edu/~greg/discrete/classnotes/exaggerate-and-simplify.pdf
  https://mediaspace.tufts.edu/media/t/1_qy6hu032
  
  
• Proof by contrapositive / contradiction / smallest counterexample:
  http://research.engineering.nyu.edu/~greg/discrete/resources.html#proof-contra
  1) https://mediaspace.tufts.edu/media/t/1_rwjwvoox
  2) https://mediaspace.tufts.edu/media/t/1_uij5x835
  3) https://mediaspace.tufts.edu/media/t/1_2i6sqbh0
  4) https://mediaspace.tufts.edu/media/t/1_9c0k7n20
  
  
• Proof by Induction:
  http://research.engineering.nyu.edu/~greg/discrete/resources.html#induction
  1) https://mediaspace.tufts.edu/media/t/1_w9jm6zny
  2) https://mediaspace.tufts.edu/media/t/1_1mc5gjm6
  3) https://mediaspace.tufts.edu/media/t/1_wn74czi1
  4) https://mediaspace.tufts.edu/media/t/1_zzcfu4pj
  
  
• Pigeonhole principle:
  http://research.engineering.nyu.edu/~greg/discrete/resources.html#pigeonhole
  https://mediaspace.tufts.edu/media/t/1_632ynap3