Mathematics Asked by user823011 on November 26, 2020

Let $f_n(x)$ be a series of continuous function on $[a,b]$. If $f_n(x)$ uniformly converge to a positive function, then $dfrac{1}{f_n(x)}rightrightarrowsdfrac{1}{f(x)}$.

The question is rather simple and I have finished it, but I have a strange question. What if we change the conditon $[a,b]$ to $(a,b)$. Then the propositon seems to be wrong (because $f_n(x)$ may not have a uniform positive lower bound). However, I stuck in giving a counterexample.

Please give me some help!

What about something like $f_n(x)=frac{n}{n+1}x$ on $(0,1)$, which converges to $f(x)=x$? Then $$ |f_n(x)-f(x)|=frac{1}{n+1}|x|leq frac{1}{n+1} $$ so $f_n to f$ uniformly. However, begin{align} left|frac{1}{f_n(x)}-frac{1}{f(x)}right|=frac{1}{x}left|frac{n+1}{n}-1right| =frac{1}{nx}. end{align} Choose $epsilon=1$. Given any $N$, choose $ngeq N$ and choose $xleqfrac{1}{n}$. Then above is $geq epsilon$.

Correct answer by zugzug on November 26, 2020

$f_n(x) = frac{1}{n} + x$ seems to be a counter-example on $(0,1)$.

The point is, to construct a function $f$ that is positive but arbitrary close to $0$ near $a$ or $b$. On a compact interval $[a, b]$ this cannot be done, as there is a positive minimum.

Answered by Peter Franek on November 26, 2020

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- Jon Church on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP