There is currently no concrete proof of the correctness of the inflationary theory. It is for this reason that there are alternative hypotheses like the Big Rebound.
Cosmological inflation removes a number of problems (they are listed in the corresponding wiki article) that existed in cosmology earlier and today it is considered the most promising by most cosmologists.
In order to finally remove the question of the correctness of the inflationary theory, it is necessary to detect the traces of relic gravitational waves. If these traces are found, the last doubts will disappear, and Starobinsky, Mukhanov and Linde will go to Stockholm for a well-deserved award.
What proof do you need? Going back 13 billion years ago and seeing with our eyes is not so easy ...
The theory describes the development of the Universe well and fairly consistently. It is also important that the universe is expanding and continues to expand. There are no prerequisites for the fact that it began to expand from a state with a finite size. By mentally turning the Universe backward in time, compressing it, we ourselves come to the theory of inflation, and then to the theory of the Big Bang (as the starting point of inflation).