If we know that a sequence $f_n$ of real-valued functions converges uniformly on the interval $[0,\epsilon]$ for every $\epsilon$ $\in (0,1)$, can we say that the sequence converges uniformly on the half-open interval $[0,1)$? If so, how to prove this fact rigorously? If not, what counter-exmple(s) can we give?
2026-04-30 04:54:34.1777524874
About Uniform Convergence
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Hint: Let $f_n(x) = x^n{}{}{}{}$.