A non-mathematician’s (programmer’s) question on infinity? 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 of 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 prove 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 needs and requires proof for the infinite case as well, instead being satisfied proving only finite case ?
Thanks for any advise!!

I personally think the main reason is to have to deal with the boundary. We cannot exactly fix the boundaries in every case. For your case it is simple when you are considering years but consider a case where you are calculating the time taken by the light to reach the other end of universe (just a real life example). You can't fix a limit in every case so you need a abstract value that works with every possible value. Hence infinity.
