Easy way to bring proof
I have a great idea.
Why don't they just put a few dead corpses on a skid, tie down the corpse, and put a long chain on the skid, leave the skid somewhere far far away from the north of the wall where the white walkers will pass by, then sit far away and wait for the corpse on the skid to turn into white walker, and then just drag the skid back along with newly turned white walker tied down on the skid.
I wonder if the white walkers need to be in freezing code air to keep it fresh, you know, like ice creams.