Here's the only argument you ever need:
The kinds of energies the LHC deals in pale in comparison to what the universe does every single second in cosmic collisions, supernovae and the like, and has been doing for billions of years. If the large hadron collider could destroy the universe, we wouldn't be able to build it in the first place on account of not existing.
Refer to this Official Resource thing here [http://public.web.cern.ch/public/en/lhc/safety-en.html] for more.