= Uniform convergence
{title2=$\|f_n-f\|_\infty\to0$}
{wiki}
= Converges uniformly
{synonym}
A <sequence> of <functions> $f_n:X\to Y$ <converges uniformly> to $f:X\to Y$, where $Y$ is a <metric space> with <metric> $d$, precisely when
$$
\forall\varepsilon>0\;\exists N\in\mathbb N_{>0}\;\forall x\in X\;\forall n\geq N:\quad d(f_n(x),f(x))<\varepsilon.
$$
For <real numbers>, use the <absolute value> $d(u,v)=|u-v|$. Compare the <universal quantification> and <existential quantification>:
$$
\begin{aligned}
\text{pointwise:}&\quad \forall x\;\forall\varepsilon>0\;\exists N\;\forall n\geq N,\\
\text{uniform:}&\quad \forall\varepsilon>0\;\exists N\;\forall x\;\forall n\geq N.
\end{aligned}
$$
The decisive change is whether $N$ can depend on $x$. <Uniform convergence> implies <pointwise convergence>.<Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl/6.1 Pointwise and uniform convergence>[Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.1.2, Definition 6.1.6 and Proposition 6.1.7]{ref}
For <functions> taking values in <real numbers> on a nonempty <domain of a function>, the <supremum error> is $E_n=\sup_{x\in X}|f_n(x)-f(x)|$, allowing $+\infty$. Then <uniform convergence> is equivalent to $E_n\to0$. The <supremum> need not be a <maximum>: pointwise strict bounds below $\varepsilon$ only give $E_n\leq\varepsilon$, so use tolerance $\varepsilon/2$ when proving this equivalence. This single error bound controls every possible input, including one chosen after seeing the index.
For $f_n(x)=x^n$ and its <pointwise limit> on $[0,1]$, the error is zero at one but approaches one as $x\to1$ from below. Consequently
$$
E_n=\sup_{x\in[0,1]}|f_n(x)-f(x)|=1
$$
for every positive <integer> $n$. This <supremum> is never attained: below one the error is $x^n<1$, and at one it is zero. Deleting the endpoint leaves the same <supremum error> on $[0,1)$. To disprove <uniform convergence> directly, choose $\varepsilon=1/2$: for any proposed $N$, take $n=N$ and $x=2^{-1/N}<1$. The error equals $1/2$, violating the strict tolerance. Thus a moving input defeats every proposed common threshold.
On the smaller <interval> $[0,a]$, for a fixed $0\leq a<1$, the <pointwise limit> is zero and
$$
E_n=\max_{x\in[0,a]}x^n=a^n\longrightarrow0.
$$
For $0<a<1$ and $0<\varepsilon<1$, the exact first successful common threshold is
$$
N(a,\varepsilon)=\left\lfloor\frac{\log\varepsilon}{\log a}\right\rfloor+1.
$$
The <logarithm> and <floor function> turn an infinite approximation statement into a finite stopping rule. At $a=0.8$, $0.8^{20}=0.01152921504606846976$ fails tolerance $0.01$, while $0.8^{21}=0.009223372036854775808$ succeeds. At $a=1/2$ and $\varepsilon=1/8$, the ratio is exactly three: $N=4$ is necessary because equality is insufficient. For $a=0$, every error is zero. The <domain of a function> determines whether one threshold can certify every input.
\Image[/powers-worst-error-and-convergence-threshold.png]
{title=<Supremum error> and common thresholds distinguish <uniform convergence> from <pointwise convergence>}
{description=Left: the error of $x^n$ remains one on $[0,1]$ and $[0,1)$, but falls below $0.01$ from $n=21$ onward on $[0,0.8]$. Right: the first successful threshold at a fixed input grows without bound near one. A common threshold $21$ covers all inputs at most $0.8$. Both vertical axes use a <logarithmic scale>.}
For another explicit stopping rule, let $h_n(x)=x/(n+1)$ on $[0,1]$. Its <uniform limit> is zero and its <supremum error> is $1/(n+1)$. For every $\varepsilon>0$, $N=\max\{1,\lceil1/\varepsilon\rceil\}$ works, using the <ceiling function>. Unlike $x^n$ on $[0,1]$, it controls even the endpoint.
The practical beauty of <uniform convergence> is that one quantitative bound can survive further operations: the <uniform limit theorem> preserves <continuity>, and <uniform convergence preserves integrals on a compact interval> controls accumulated error. <Differentiation> needs additional hypotheses, such as <uniform derivative convergence with an anchored value>. These results distinguish an approximation that looks accurate at selected inputs from one that justifies replacing a whole <function> in a calculation.
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