Apple Mail is not showing Sent Items in Gmail. The mails sent from Apple Mail are not in Gmail.
Ensure that in your mail settings you have:
IMAP Path Prefix: [Gmail]
Under the Mailbox Behaviors tab in Accounts settings you have checked:
√ Store sent messages on the server Delete sent messages when: Never
In the Sidebar of Mail, there should be grey subheadings for your accounts. Under the Grey subheading for your gmail account, you will find folders with titles of your gmail tags (e.g.
Starred, etc.) If you see one there called
Sent Mail, select it, and select this menu item:
Use This Mailbox For >