You can download the mails from your Gmail and import it to your mac mails
- create a label like "September mails"
- Move all september mails to your label
- Go to your account settings

Data tool > Select data to download

Choose your label and download the mails

The data is exported in standard, widely supported formats—MBOX for e-mail. Most e-mail clients and services can read these files. From Apple support:
Mail (Mavericks): Import mailboxes
Mail can import messages from other email apps or from mailboxes that you exported using Mail.
- Choose File > Import Mailboxes.
Select the email app you want to import messages from.
If you’re importing a mailbox exported from Mail, or a folder (such as [email protected]) from the ~/Library/Mail/V2 folder on another Mac, choose Apple Mail.Follow the instructions that appear.
Make sure the app you’re importing from is active, because some of
the options require this.
If you’re importing messages from a Windows or UNIX computer, select
“Files in mbox format,” then locate the folder containing the files.
If the email app you’re switching from isn’t listed, click Cancel.
Consult the documentation for the other app to learn how to export
messages in mbox format (most email apps can). Then import the mbox
files.
Review and reorganize the imported messages.
The messages are located in a new mailbox called “Import” in the
sidebar. To reorganize the messages, create new mailboxes and drag
folders and messages from the Import mailbox to other mailboxes. When
you’re done, you can delete the Import mailbox.
Hope this helps!