When the update process is complete, iTunes may not give you any notification. Sometimes, iTunes just automatically disconnects your iPhone from the software and then connects it again. This happens quickly, and you may not even notice it.
Alternatively, you may see a notification that iTunes is going to reboot your iPhone. Let this process run.
Once the update process is complete, iTunes will tell you that your iPhone is running the current version of the iPhone software. You'll see this information on the iPhone Summary screen.
To verify that your iPhone software is up to date, look at the top of the iPhone summary screen. You'll see some general information about your iPhone, including which version of the iPhone software it's running. This version should be the same as the software you just downloaded and installed.
Before you disconnect your iPhone from your computer, make sure that iTunes is not backing it up or syncing it again. When iTunes is syncing, your iPhone screen will display a large message that says "Sync in Progress." You can also check the iTunes screen; you'll see a message at the top of the screen that tells you if the backup and syncing progress has finished.
Congratulations, your iPhone has been updated!