Gmail account works on one Mac, not on another

Question

I have a late 2009 iMac and an early-model MacBook Air. Both were running the latest Mountain Lion; I updated both to Mavericks, so they are both running OS X 10.9 and Mail 7.0.
Before the upgrade, both Macs had a mail account. (It's the mail service for my ISP, but it's really just Gmail.) Mail worked equally well on both.
After the upgrade, Mail continued to work perfectly on the iMac, but it failed completely on the MBA. I get a warnint triangle next to my Inbox. When I click on it, Connection Doctor tells me that I have a good internet connection, but that it could not connect to either the Gmail SMTP server or the IMAP server.
I've checked every detail of the account settings from Mail preferences; they are identical. I've deleted and recreated the account on the MBA multiple times, from both the Mail preferences accounts tab and the System Preferences Internet Accounts tab. Nothing helps. Mail on the MBA is totally unable to connect to the Gmail servers, no matter what I do.
Here are some of the details:
Account Information:
Enable this account: checked
Account Type: Google IMAP
Description: Tellink IMAP
Email Address: ([email protected] (my ISP))
Full Name: (my name)
Incoming Mail Server: smtp.gmail.com
User Name: (same as my email address)
Password: (my Gmail password)
Outgoing Mail Server (SMTP): Tellink (Offline)
Use only this server: Checked
TLS Certificate: None
Edit SMTP Server List:
Description: Tellink
Server Name: smtp.gmail.com
In Use By Account: Tellink IMAP
Use custom port: 465
Use Secure Sockets Layer (SSL)
Password: (my Gmail password)
Advanced:
IMAP Path Prefix: (empty)
Port: 993
Use SSL: checked
Use IDLE command if the server supports it: checked

Answer 1

This was fixed by the upgrade to 10.9.1.