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I'd love your help with understanding what led me to conclude that $\arctan(nx)$ converges uniformly for $x \in (0, \infty)$

I wanted to check that $\lim_{n \to \infty} \sup |f_n(x)-f(x)|$ is 0 where $f(x)=\lim_{n \to \infty} f_n (x)$.

$f(x)=\frac {\pi}{2}$.

I interpret $\lim_{n \to \infty} \sup |f_n(x)-f(x)|$ as: for every $x_0$ in my interval, compute $\lim_{n\to \infty} |f_n(x_0)-f(x)|$=$L_{x_0}$ and then take the $\sup (L_{x_0})$ for all $x_0$ in my range. Am I right?

So, In my case I have $f(x)=\lim \arctan (nx)= \frac {\pi}{2}$

$r_n(x)=f_n(x)-f(x)=\frac {\pi}{2}-\arctan(nx)$

So for every $x \in (0, \infty)$ , small as can be, when I'll compute $\lim \frac {\pi}{2}-\arctan(nx) =0$ when $n \to \infty$ so $\sup (L_{x_0})=0$ since all the values in this set is $0$. What do I get wrong here? what is the correct defenition for$\lim_{n \to \infty} \sup |f_n(x)-f(x)|$? I feel like I got it all wrong.

Thanks guys!

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2  
You've got the definition backwards. For each $n$, compute $\sup_{x \in [0,\infty)} |f_n(x) - f(x)|$. Call this number $s_n$. You have uniform convergence only if $\lim_{n \to \infty} s_n = 0$. –  Nate Eldredge Jan 28 '12 at 14:19
1  
You are confusing $\lim_{n \to + \infty} \sup_x |f_n (x) - f(x)|$ and $\sup_x \lim_{n \to + \infty} |f_n (x) - f(x)|$. They are not the same, as this example shows. –  D. Thomine Jan 28 '12 at 14:23

3 Answers 3

up vote 3 down vote accepted

Set $f_n(x)=\arctan(nx)$. Then $\{f_n\}$ converges pointwise to $f(x)={\pi\over2}$ on $(0,\infty)$.

If $x_n={1\over n}$, then $f_n(x_n)=\arctan(1)={\pi\over4}$. So, the sequence $\{f_n\}$ does not converge uniformly to $f$ on $(0,\infty)$.

You have to compute the $\sup$ for each fixed value of $n$ first, and then take the limit. From the above, we have for any $n$:$$\sup\limits_{x\in(0,\infty)}|f_n(x)-f(x)|\ge {\pi\over 2}-{\pi\over 4}={\pi\over 4 }.$$ From this, $$\lim\limits_{n\rightarrow\infty} \sup\limits_{x\in(0,\infty)}|f_n(x)-f(x)|\ne 0.$$


A picture may help:

enter image description here

Note that, in fact, $\sup\limits_{x\in(0,\infty)}|f_n(x)-f(x)|={\pi\over2}$ for all $n$.

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Am I allowed to choose $x_n={\pi\over 4n}$ since it's limit when $n \to \infty$ is 0 and 0 is not in our range? –  Jozef Jan 28 '12 at 14:48
    
@Jozef: It matters only that $\pi/4n$ is in your domain. The fact that $\pi/4n \to 0$ indicates that the reason the functions $f_n$ do not converge uniformly to $f$ is that near $0$ the uniform convergence is lost. –  Beni Bogosel Jan 28 '12 at 14:51
    
Thanks a lot David. –  Jozef Jan 28 '12 at 14:54

Uniform convergence means that the distance between $f_n(x)$ and $f(x)$ becomes small for all $x$, at the same time. To emphasisze here are the two definitions:

  • $f_n$ converges to $f$ pointwise if for every $x$ and every $\varepsilon >0$ there exists $n_0$ such that $|f(x)-f_n(x)|<\varepsilon$ for all $n \geq n_0$.

  • $f_n$ converges uniformly to $f$ if for every $\varepsilon >0$ there exists $n_0$ such that $|f(x)-f_n(x)|<\varepsilon$ for all $n \geq n_0$ and for all $x$.

In your case, the problem is that every function $f_n$ can be extended continuously by $f_n(0)=0$, so near $0$ the functions $f_n$ cannot approach $\pi/2$ simultaneously for small $x$. If your functions would have been defined on $(\delta,\infty)$ with $\delta>0$ then the convergence would have been uniform.

Another way to see that $f_n$ does not converge uniformly to $f$ is to consider $f_n :[0,\infty)$, extended by continuity. It is relatively easy to prove that if $f_n$ converges uniformly to $f$ and all $f_n$ are continuous then $f$ is also continuous. But the limit of your functions $f_n(x)=\arctan (nx)$ is

$$ f(x)=\begin{cases}\frac{\pi}{2},& x>0 \\ 0,& x=0\end{cases}$$

which is not continuous.

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Found myself attempting the same question as original poster, found this answer to be most explanatory! ty. –  Faust7 May 13 '13 at 22:40

Here is another reason to see that the convergence is not uniform. The functions $\arctan(nx)$ converge pointwise to the function that is 0 at 1 and $\pi/2$ for any positive number. Each function is continuous, but the limit is not. Hence, the convergence cannot be uniform.

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