What exactly is your question? A sequence may converge uniformly on many sets. In fact, one of the first things you should have learned about "uniform convergence" is "if a sequence converges uniformly on a set, then it converges uniformly on any subset of that set". Another is "if a sequence converges on a compact set, then it converges uniformly on that set".

A standard way of proving that a sequence of functions converges uniformly on (0, 1) is to look at the closure of that set, [0, 1]. Since that is both closed and bounded it is compact so you only have to prove the sequence converges on it to know that it converges uniformly. Then, since (0,1) is a subset the sequence also converges uniformly on (0,1).