How do I make sure that OS X Mavericks is successfully installed?
Click the  in the top left of your screen and select "About This Mac". If it says "Version 10.9.x" then you have successfully upgraded to Mavericks!



The installer app should have removed itself, but if it didn't you can go ahead and delete it manually.

And in the App Store it should show you a "Download" button instead of a greyed out "Installed" button. This is so you can download it again to reinstall OS X or install on another Mac. The App Store is not the place to check what version of OS X is installed, only to download apps and new versions of OS X.