IPhone 5 won't connect to Mac?

I've just bought an iPhone 5 on eBay.

I've previously had a 4s so already have iTunes installed and ready.

I thought it would be a simple case of plugging in the new iPhone and hitting the restore button and setting up new phone from back up.

Since plugging it into my Mac, nothing is happening! It won't even charge, just doesn't connect at all yet it charges with the same lead from the wall.

I've looked on YouTube and followed the instructions to press and hold power buttons (top and bottom ones) and I get the screen come up on the phone that says 'connect to iTunes' then still nothing comes up on screen on the Mac.

Please don't tell me I've bought a fake. This is what I'm dreading. I asked for proof it was genuine and he showed me an email from apple confirming that this was a replacement for a faulty one he returned. The email contained the imei number which matches so thought I'd done my checks.

Please help. Spend hours messing around with it. I'm not very tech savvy as you can probably tell so simple instructions will be appreciated.

Need to reboot

Did you update itunes to the latest update?

Here are a few things to try:
Delete itunes and redownload it
Sign out of your itunes account and then plug phone in, or vice versa
Hold home button down while plugging phone into the computer until the info pops up
Try holding the home and power button to reset it while it's plugged into the computer
Make sure your itunes is running on the latest version
Go into the settings of the phone and see if it can recognize the computer
Plug phone in and if it doesn't pop up on the computer while itunes is running, close itunes while phone is still plugged in and see if it asks you if you want to restore phone (thats what worked for me)
Delete itunes and then plug phone in and see if the computer recognizes the phone is plugged in. If it does it will either ask you if you want to download itunes, or import pictures. If it doesn't recognize it right off the bat, go into the documents part of your computer (or wherever it shows what you have plugged in) and click on the iphone that shows up
Try plugging the phone into the computer while the phone is turned off, and see if it turns on when you plug it in
Restart computer
Turn phone of and turn it back on bu holding both buttons or vice versa
You may need to authorize your computer, which you can do from the apple website (it will give instrutions for which kind of computer you have)
If none of these work, sigh into itunes from the app on the phone and see if it will let you sync your old things from the other phone (this does work sometimes)
If you used icloud, log into icloud on your computer or on the phone and see if it allows you to sync it there
Make sure your new sim is in the phone when you plug it into the computer, if it it, try plugging it in when the sim card isn't in the computer. You can also try the previos suggestions after switching the sim in or out

If none of these work, call Apple and see if they can verify that the phone is real or not. You could also take it to an Apple store and tell them that you don't know why it isn't working. They may fix it, or replace. You may not want to tell them that the phone wasn't bought from them, because they may not fix it or replace it.

Sometimes Macs are tricky when it comes to these things, which sounds weird since they are also made by Apple, but it's true. I have this problem sometimes and it just means that you need to retry it over and over until it works. It just takes the right combination of doing things or pressing buttons in the right order.