Integration and Uniform Convergence

Real Analysis · Axiom Academy

EXAMPLE Integration and Uniform Convergence When does ? Uniform convergence is the key. Suppose uniformly on [a,b] . Show that we may interchange the limit and the integral: Then test it on the concrete sequence f_n(x) = x^n on [0,1] — and see what goes wrong when convergence is only pointwise . Uniform: fₙ trapped in an ε-tube around f Once n is large, f_n lies within of f everywhere at once , so the area between them is at most . Pointwise only: a spike of fixed area Each f_n peaks higher and narrower. At every fixed x , , yet the enclosed area does not shrink — there is no single -tube to trap it. Nice work. You proved the interchange theorem from a one-line estimate, confirmed it on f_n(x)=x^n , and saw why pointwise convergence alone is not enough. The whole proof is one estimate: . Uniform convergence is what kills it: drags the right-hand side — and so the gap — to 0 . Why "uniform" matters: uniform means the sup is independent of x — one bound works across the whole interval, so it controls the area all at once. Pointwise alone fails: for on [0,1] , pointwise yet . The spike's area never vanishes. Sufficient, not necessary: x^n on [0,1] is not uniformly convergent ( ), yet its interchange still works — the theorem gives a guarantee, not the only way.

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