In https://en.wikipedia.org/wiki/Uniform_convergence it's given that

A sequence ${f_n}$ of functions converges uniformly to a limiting function $f$ if the speed of convergence of $f_n(x)$ to $f(x)$ does not depend on $x$.

Now the definition for uniform convergence is

Suppose $S$ is a set and $f_n : S → R$ is a real-valued function for every natural number $n$. We say that the sequence $(f_n)_{n \in > \mathbb{N}}$ is uniformly convergent with limit $f : S → R$ if for every $ε > 0$, there exists a natural number $N$ such that for all $x ∈ S$ and all $n ≥ N$ we have $| f n ( x ) − f ( x ) | < ε$.

Now where does it require for the speed of convergence to not depend on $x$?

Does for example $f_n=\frac{1}{\sqrt{n}}$ converge uniformly even if it certainly does not have the same speed of convergence for all $x$?

  • $\begingroup$ Uniform convergence pertains to a sequence of functions. $1/\sqrt{x}$ is not a sequence of functions, it's just one function. You may have misunderstood something... $\endgroup$ – Thompson Sep 17 '16 at 16:50
  • $\begingroup$ The speed of convergence indicator is the number $N$. Well if we look at the definition, $N$ does not depend on $x$ therefore the speed of convergence is independent of $x$ as well. $\endgroup$ – John11 Sep 17 '16 at 16:52
  • $\begingroup$ @John11 But perhaps it's that the "speed of convergence" does not refer to how much there's between successive $f_k$s but rather that for $n \gt N$ the convergence does not "disappear" in some $x$s but that there's constant convergence (although not necessarily at the same speed? Or necessarily at the same speed?)? $\endgroup$ – mavavilj Sep 17 '16 at 16:54

"Speed of convergence" is a somewhat fuzzy informal term for how fast the necessary $N$ increases as $\varepsilon$ drops towards $0$.

With pointwise convergence each $x$ can have its own dependence between $\varepsilon$ and $N$. On the other hand, uniform convergence requires that you can give an $N$ based on only what $\varepsilon$ is, and that $N$ then has to work for every $x$.

In other words there is a function from $\varepsilon$ to $N$, and this function (intuitively encoding the "speed of convergence") does not depend on $x$.

Uniform convergence does allow the sequence to converge faster than this common function for some particular $x$, so it would perhaps be more precise to say that the "speed of convergence" is bounded rather than to say that it is independent of $x$.


OK I see. I agree with you. Don't believe everything on wikipedia. There might not be a way to say it that is both succinct and completely true.

Uniform convergence means that "the rate of convergence is bounded below independently of $x$" or something.


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