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Re: why won't my iPhone 5 update my emails onto the phone?
Welcome to iMore! Go to Settings > Mail, Contacts, Calendars > Fetch New Data, and make sure that your email accounts are properly set. They should be set for Push (if available) or Fetch. Also make sure that you have a time increment set under Fetch on that same screen (i.e. every 15 minutes, ever 30 minutes, etc.). I recommend setting it for a time increment you're comfortable with; just don't set it for "Manually" in your case.
While you're at it, make sure your phone isn't in low power mode. You'll know because your battery indicator will be yellow instead of green. If low power mode is enabled to go Settings > Battery, and toggle it off.
If you're still having problems, go to Settings > Mail, Contacts, Calendars, then select your email account and tap Delete Account. Once that's finished add it back in.