iPhone 5s internet not working
First, go to Settings > General > About, and wait to see if the "Update Carrier Settings?" Dialogue box appears. 

Then, hard reset your phone by holding down the Home and Power button until you see the Apple logo. 

If that doesn't resolve your connectivity issue:
Go to Settings > General > Reset, and tap "Reset Network Settings". Enter your passcode if you have one set. 

Tap "Reset Network Settings" again. This will cause your phone to restart, and you will lose all stored Wi-Fi passwords. 

However, it usually resolves problems like this. Back up your phone first if you aren't sure you remember all of your Wi-Fi passwords.