So everybody here probably knows if numerical testings support a conjecture, that conjecture isn't necessarily true for larger numbers. In fact,there have been many times where a conjecture was proved to be wrong by large counterexamples. But is there any conjecture that suggests if a conjecture is true for all $x<n$ for some $n$, that conjecture holds for all values of $x$?
Please note that I'm not looking for just one specific conjecture that's proven to be true for all $x>n$ for some $n$ and we just need to prove it for all $x<n$ such as Goldbach's weak conjecture(I know it's been proved that it's true for all odd numbers but if I recall correctly, someone proved it's true after a very large number but we couldn't test all the odd numbers less than that number). I'm looking for something that could be applied to any conjecture(that could be numerically tested) or something that at least could be used for a large variety of conjectures.
Thanks in advance!