I apologize for my total ignorance in the sphere of mathematics and the possibly very silly question I'm about to ask. My mathematical knowledge level is quite limited (pretty much finished with some slightly more advanced stuff then grade 12) so please if possible limit too much terminology to about that level 12 math. Again I don't mean to offend anyone & I'm sorry if the following sounds like a joke but I am genuinely interested and cannot quite grasp the reason for it.
I've been curious for quite sometime now, What is the significance for a mathematics to frequently require proof for both finite & infinite cases of theorems ? Why isn't it satisfactory to proof any theorem for a reasonably high finite x (whatever x is - be it set of some numbers) ? The reason why I'm asking is that in real-life applications (not talking about software application but life applications like count a bag of money or something like that) there is likely never need to deal with infinite of anything really - it might be a very high quantity but never infinite. So why does mathematics need and requires proof for infinite case as well instead being satisfied if finite case ?
Thanks for any advise!!