To assure all folders are synchronized (if using multiple devices or computers) make sure you are using the IMAP configuration.
If your sent emails are not showing up on your device perform the following steps.
1. Launch the Settings app
2. Choose Mail Contacts and Calendars
3. Select the account with which you are having trouble syncing the Sent folder
4. Under IMAP tap on your Account. (NOTE: If it says POP, then you have to delete the account and re-add it as an IMAP account).
5. Choose Advanced at the bottom
6. Under Mailbox Behaviors, tap on “Sent Mailbox”
7. Ensure that the folder called “Sent” is checked off under the “On the Server” section
To save the settings:
Click the “Advanced” button on the top left.
Click on “Account Info”.
Click “DONE” (do not click Cancel).