It's actually not an unreasonable question. One simple answer might be that "we don't" -- physics relies on the assumption that the laws of physics are identical regardless of where you are in the Universe, and this seems to work as an assumption, but it's not really provable.
For your specific question, a sudden "stop" of the expansion rate can be reasonably safely ruled out. There is a general principle that most things happen fairly smoothly, so a sudden stop can be ruled out on those grounds, although again it's more a principle than a fact. On the other hand, the data we have on expansion comes across a whole set of effective age ranges from just a few tens of millions of years to almost the start of the Universe, so we can be confident that the Universe was expanding over almost all of its life so far. Since in that time the expansion rate has also appeared to increase, it seems unlikely that suddenly in the last couple of million years that picture has changed to no expansion at all. In order to achieve this something would have to happen at every point in space to reverse the expansion rate. Whatever that thing it you can bet its consequences would be fairly spectacular and not go unnoticed even at local scales.
All the same, it's not really certainty. But the next best thing, "overwhelming balance of probability".