That’s a really great question. The answer is that mathematicians keep their statements general when trying to prove things. Another commenter gave a bunch of examples as to different techniques a mathematician might use, but I think giving an example of a very simple general proof might make things more clear.
Say we wanted to prove that an even number plus 1 is an odd number. This is a fact that we all intuitively know is true, but how do we know it’s true? We haven’t tested every single even number in existence to see that itself plus 1 is odd, so how do we know it is true for all even numbers in existence?
The answer lies in the definitions for what is an even number and what is an odd number. We say that a number is even if it can be written in the form 2n, where n is some integer, and we say that a number is odd if it can be written as 2n+1. For any number in existence, we can tell if it’s even or odd by coming back to these formulas.
So let’s say we have some even number. Because we know it’s even, we know we can write it as 2n, where n is an integer. Adding 1 to it gives 2n+1. This is, by definition, an odd number. Because we didn’t restrict at the beginning which even number we started with, we proved the fact for all even numbers, in one fell swoop.