Suppose $\{f_n\}$ is a sequence of complex functions and $|f_n(x)-f(x)|\to 0$ for all $x$. If we put "for all $x$" behind the $|f_n(x)-f(x)|\to 0$, does it show that the convergence is uniformly convergence?
2026-02-23 08:29:34.1771835374
a notation for convergeence.
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No. The sentences
express the same fact. It's just a matter of rewriting the sentence in an equivalent form, which is a feature of English and many other languages.
You pass from pointwise to uniform convergence when you interchange the quantifiers in the $\epsilon$-$\delta$ definition of the convergence, namely you replace $\forall \epsilon \forall x \exists \delta \cdots$ with $\forall \epsilon \exists \delta \forall x \cdots$.