Delete definitely a message from Mail I am subscribed to a newsletter, everyday I get emails but once I read them I delete them. After sending them to the trash and emptying it, next day I receive the usual email, but the previous email is also shown below the new email. The problem happens with my gmail and hotmail accounts.

How can I set things up so that when I delete a mail from my Gmail Inbox, it also gets deleted on the servers and other devices?

I don't want them to be somewhere around I don't need them. Every time I receive an email from the newsletter I got a list of all the emails dating back to 3 or more months. I remember I delete them one by one and then emptying the trash but no luck next day there were all of them as part of the "conversation".
There is a several hundred word long tutorial, 

Full Step-by-Step Tutorial  taken from here, that I have used to solve this problem so in addition to the author testing it, I too have things fully working.

Here are the steps from the article as well as details from the two key steps.

Step 1: Configure Apple Mail preferences  

Step 2: Assign mailbox functions

Step 3: Set the [Gmail]/ prefix on custom labels

Step 4: Designate which labels show in IMAP

Since this is the most important step, here are the details


  While you’re in the Labels tab, uncheck the “Show in IMAP” box for All
  Mail because you don’t want that showing up in Apple Mail anymore.
  This is important to ensure the delete button sends messages to the
  Trash in Mail.app. The “Show in IMAP” box will determine which labels
  appear in Apple Mail. Feel free to uncheck the “Show in IMAP” box for
  any other labels while you’re at it. Changes are saved automatically.


Step 5: Forwarding and POP/IMAP

This also is a key step where google allows you to delete a message once it's removed from tha “last visible IMAP folder.” By hiding the All Mail label in IMAP, there should only be one “visible” copy of your message, so when you delete the message in Mail, it will now move the message to the Trash.



Step 6: Restart Apple Mail

Step 7: Test it out

I've tested this on Lion 10.7.5 and works perfectly.
All images are from Jeff Schuette as is the quoted text explaining how google can be set up to work like all the rest of the Mail servers to delete messages when you trash them.