Restoring a new iphone from an older backup?

My mom just got the iphone 6s and her old one was a 5s i believe. I did a backup of the old phone, and now trying to up that on the new phone. When I first tried it said it cant be restored because the software on the iphone is too old. (to restore this iphone from this backup, you just first set up a new iphone and restore the software to the lastest version.) So i started a new iphone and when i click the restore back up the one i need isnt there. But when i go to preferences I can see the backup i need so it does exist. Idk what it wants me to do. Am I suppose to update the old iphone and then back it up? Im not even sure if it has a new update. And we dont even have icloud set up but if anything i guess thats the next option.

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