The included iOS email client does what you want. If you configure as Exchange the emails are pushed to you instead of pulled. Use m.google.com as the Exchange

The included iOS email client does what you want. If you configure as Exchange the emails are pushed to you instead of pulled. Use m.google.com as the Exchange server. Also, the trash icon doesn't actually delete anything. It sends the message to your archive. This is the preferred way because it is push and also syncs the contact list with Google. Alternatively, if you configure the included client as Gmail it will use an archive icon instead of the trash can, but it is not push enabled. You will have to pull every 15 minutes. You wouldn't get the contact syncing. You may need to recreate your email account to get the behavior you want if you you made it as IMAP before Exchange and Gmail were options.