Let f and f_{n} :[0, 1] -- R be continuous functions on [0, 1], for all n natural numbers. Suppose that the sequence f_{n} converges uniformly to f, on each compact subinterval [0, a], with a<1.
In addition, suppose that f_{n}(1) converges to f(1) as n converge to infinity (or, in particular, suppose that f_{n}(1)=f(1), for all natural number n).
The question is if from these hypothesis, necessarily we have that the sequence (f_{n}) converges uniformly on the whole interval [0, 1] ?
If the answer is negative, then I would greatly appreciate a counterexample.
Thank you very much in advance for your help.
George
