The headline is certainly overblown, but there probably errors out there. The interesting question is where we should be looking for them. All of the major re

The headline is certainly overblown, but there probably errors out there. The interesting question is where we should be looking for them. All of the major results in an undergraduate or introductory graduate textbook have been studied often and thoroughly enough that we can be reasonably confident there are no errors in their proofs. There are errors in very new or very obscure material, but those tend to get corrected as more people start paying attention to them. It therefore seems like the search should therefore be concentrated in recent results that are becoming more useful but still aren't standard. On the other hand, we do need to build a standard library of results that can be used to prove those newer results. You can't do much in analysis without basic results on metric spaces, for instance.