cosmology - How do we know that there is no border at the end of "infinite" space?

We do not know what lies beyond the limits of our instruments, by definition.

For all we know, there could be a big sign saying "This area under construction."

However, that doesn't seem very probable.

The most probable answer is "more of the same". After all, there is no reason the bit of the universe we can see should be different from other places.

If you keep adding "more of the same" to the universe, you get the infinite universe you have seen described.

It is just a theory, but it seems like a pretty probable one.

It is also easier to do calculations on. Having a border introduces all kinds of problems like galaxies near the border only getting gravitational forces from one side only. With an infinite universe these forces come from all directions and cancel each other.

It is still just a theory, but it is one that is (relatively) easy to work with.