Mail Does Not Sync Automatically

I've configured several e-mail accounts in Mail. Three of them are Microsoft Outlook accounts and two are Gmail accounts. For each account I've set Mail to:

(a) Fetch new messages as they arrive
(b) Download messages from all time

There is no other special setting.However, Mail never seems to fetch anything automatically as they arrive. In fact, I always receive my e-mail on my phone long before they show up in my inbox. As a test, for most of yesterday and today I waited about an hour after messages arrived in my phone, and each time I found that until I had manually synced the account, the messages did not appear in Mail.

Some additional information : I am certain that my Internet connection is working, because I get Skype messages in real time; I am sure my computer has the correct drivers, because not only did I install everything from the manufacturer's site, I even ensured that there were no errors in Device Manager; my computer has plenty of RAM and CPU horsepower. There is no reason for Mail not to sync except if it has a bug.