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It doesn't support 5 GHz.

It does, indeed. So did the iPhone 5.

This model just seems to have more trouble searching out in that spectrum. Side-by-side the 5 discovers and connects almost immediately after cycling WiFi off and on again, which the 5S sometimes requires several cycles. Maybe it is time to take the phone in for a replacement.
 
I have isolated the 2 & 5 Ghz band on the router with different APs ... and then I just select the 5Ghz manually if I want ...

Also helps keep different DNS settings on iphone for the same outbound ISP if I want ;)
 
Same here, running exclusively on 5GHz n at home, and the iPhone 5 and 5s connect fine to it. Neither have issues searching for the 5GHz networks, either.
 
Don't forget that the 5ghz band has weaker wall penetration than 2.4ghz. In can use 5hz in the house but to reach the garden I have to hook onto the 2.4ghz.
 
I have a 5, and wife has 5S. we can be in same room, I'll connect to 5ghz and she will connect to 2.4ghz. Happens consistently, and I have Airport Extreme 5th gen.
 
Don't forget that the 5ghz band has weaker wall penetration than 2.4ghz. In can use 5hz in the house but to reach the garden I have to hook onto the 2.4ghz.

It's useful though, in neighborhoods where everyone has a router and is blasting away at 2.4GHz in 40Mhz-bandwidth mode, pretty much choking up the spectrum and interfering with each other. Where I live, there's easily a dozen wireless access points visible in the 2.4Ghz band, all named "linksys" or "netgear" and overlapping each other leaving not a single clear channel. Meanwhile, my routers are the only ones on a very spacious, empty 5Ghz. I also seem to be the only one not complaining that Wifi is too slow.
 
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