Uniform Convergence

Real Analysis · Axiom Academy

A stronger convergence: one single N pulls the whole graph of f_n inside an -tube around f — all points at once. The clean way to say " uniformly on S " measures the worst-case gap over the whole domain — the supremum distance — and demands it go to zero. The largest gap over all x must vanish Equivalently: one N from alone, then for every x Geometrically: draw the band around the limit. Uniform convergence means that from some index N on, the entire graph of f_n lies inside that tube — not just point by point, but all at once. The two notions differ only in the order of the quantifiers — but that order is everything. Pointwise lets N chase each point; uniform forbids it. For each x separately, some N (allowed to depend on x and ) makes for . Different points may need ever-larger N . A single N (depending only on ) works for all x at once. The same N is enough everywhere. 3. A Uniform Example: f_n(x)=x/n Take f_n(x)=x/n on [0,1] . Pointwise the limit is f(x)=0 . Because x/n is increasing in x , the worst-case gap sits at the right endpoint x=1 : That supremum is 1/n , and . Given any , pick ; then for all and all we get . The threshold N never looked at x — convergence is uniform. 4. Pointwise but Not Uniform: f_n(x)=x^n Now take f_n(x)=x^n on [0,1] . It converges pointwise, but the limit jumps: At each fixed we have , while 1^n=1 holds the endpoint up. So the limit f is discontinuous at x=1 — already a warning sign, since the f_n are all continuous.

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