what this neglects to mention is that in order to do this, your old sim needs to be a micro/nano sim...
if your old sim isn't a micro or nano one, then DON'T FORCE it in, you will break the sim and the phone and that'll be costly
your best bet, honestly, is the age old paper and pen method of copying them over one at a time. either write them onto a piece of paper and then copy them to your iphone or just type them straight to your phone (i prefer the paper method as if you get distracted you can tick off where you're up to and just carry on but different strokes for different folks)