Uniform Convergence on [0,a] for a < 1

Real Analysis · Axiom Academy

EXAMPLE Uniform Convergence on [0,a] for Show that f_n(x) = x^n converges uniformly to 0 on [0,a] — and find an explicit N . Fix a constant with and consider the sequence of functions f_n(x) = x^n on the closed interval [0,a] . On the full interval [0,1] this sequence does not converge uniformly. Show that pulling the right endpoint back to fixes that: prove uniformly on [0,a] , and produce an explicit N that works for a given . Static figure (no animation): each curve is x^n restricted to [0,a] . The dot at x=a marks the largest value on that interval, . As n grows the dots march down toward 0 — that downward march is uniform convergence. (On [0,1] the curves would all reach height 1 at x=1 , so the worst case never shrinks.) Nice work — you proved uniform convergence on [0,a] and pinned down an explicit N . The whole proof turned on one move: control the worst case over the interval. Uniform vs. pointwise: the N depends only on , never on x — one N must work for every point at once. The supremum is the whole game: because x^n is increasing, so its largest value sits at the right endpoint x=a . The interval matters: on [0,a] with we get , so uniformly. Why [0,1] fails: there for every n (the height near x=1 never drops), so it does not and convergence is not uniform.

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