Yes, we will exist... and the people from that time will laugh at those of us who believed in iminent apocalypse, just like we laugh of those who believed in it in the past...
It's interesting how there is always some popular apocalyptical theory. First the biblical version, then the fear of endless global wars, then the fear of a nuclear confrontation, and now the "trendy apocalypse" is related to global warming. Of course there are also other versions, like the one that claims that when oil runs out we will all fight to death. Another one states the same thing, only replacing oil with water.
They are of course all wrong, because they take a couple of facts and exaggerate to the point of nonsense. In 200 years we will be alive and most probably well.