Mac Software Update Permissions Problem

I ran into a problem installing some software updates on a Mac and thought I'd post the solution here in case anyone else runs into the same problem.

I was logged in as one user, and ran Software Update, but didn't install the updates. I then created a new user account and logged in as this new user to actually install the updates.

But Software Update refused to install some of them, with this error: "You do not have appropriate access privileges".

Software Update downloads updates into /Library/Updates and the updates were there, but it seems that because they belonged to someone else, Software Update was having trouble. Deleting the updates from this folder and running Software Update again fixed the problem (the updates were downloaded again the next time I ran Software Update).