When was the last time you restarted your Mac Air? Try that first.
Then Sign Out and then back in to LR. You might need your serial number again, I'm not sure, so it would be wise to have it handy.
Download the update patch from the Keeping Lightroom up-to-date page at http://blogs.adobe.com/lightroomjournal/2013/06/keeping-lightroom-up-to-date.html#lrcc2015. Adobe's in app updates are notoriously fickle. Even this page is incredibly hard to find on the Adobe site, so I keep it bookmarked.