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Go into Settings then Mail,Contacts,Calendar
Click on Fetch New Data
Make sure Push is turned on (showing green)
Select your email account listed and make sure it says PUSH as well.
What email provider are you trying to check?
Please do not give negative ratings while we work on the issue. I am trying to replicate the issue that you are having on my end
Log into the webmail on your computer here: http://mail.yahoo.com
Move all of your inbox emails into another folder
Open your iPhone inbox now and the emails should disappear
Move the emails back into your Inbox on your computer and the emails should download again on your iPhone.
Check to see if the Push works now.
Has it ever worked with your phone?
Let's try this...
Delete the Yahoo account out of your phone
Go into Settings then iCloud and sign out
Restart your phone
Go back into Settings then iCloud and sign back in
Add your email account again
Go back into the notifications and make sure it is set to Push
Send yourself an email to see if it is pushed
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