I received a hand-me-down iPhone 5 and so far it's been working great. However, several apps I try to install off the app store tell me I need to upgrade to a later iOS version. For some reason I can't upgrade. I tried searching google, various forums, etc but no answer seems to match the problem I'm having. Here are some details on what's happening:
I go to Settings -> General -> Software Update and I see this screen:
I tap "Download and Install", and it shows this screen for about a second:
After this screen shows for about a second it just goes back to the first screen. I've tried it several times with no avail. I am connected to Wi-Fi and the iPhone is plugged in when I do this. I've also rebooted the phone several times.
In case it helps, here is what I see when I go to Settings -> General -> About:
Does anyone know to resolve this issue? I can provide additional information if necessary. Thank you!
I go to Settings -> General -> Software Update and I see this screen:

I tap "Download and Install", and it shows this screen for about a second:

After this screen shows for about a second it just goes back to the first screen. I've tried it several times with no avail. I am connected to Wi-Fi and the iPhone is plugged in when I do this. I've also rebooted the phone several times.
In case it helps, here is what I see when I go to Settings -> General -> About:


Does anyone know to resolve this issue? I can provide additional information if necessary. Thank you!