Make sure you hard reset into DFU mode again before continuing, and try this method a few times using different cables or computers to make sure it’s not an external hardware problem. [br]
To get into DFU mode on the iPhone 5 press and hold both the Home and Power button for 10 seconds, and then release the Power button, but '''continue''' to hold the Home button for 10 more seconds.[br]
Now the phone should be in DFU mode and you can install the latest OS available for your device in iTunes. [br]
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If all else fails take it in to a service/repair shop for a technician to have a look at.