# Thread:

1. ## Uniform convergence

I am asked to show that $f_n (x) = x^n$ is uniformly convergent in $(0, 1)$ but not in $[0, 1]$.

I proceed by computing the pointwise convergence to $f(x) = 0$ for $x \in [0, 1)$ and $f(x) = 1$ for $x = 1$.

Then I compute for $x \in [0, 1)$:

$\vert f_n(x) - f(x) \vert < \epsilon \Leftrightarrow \vert x^n \vert < \epsilon \Leftrightarrow x < (\epsilon)^{1/n}$

This is not fullfilled for all $x \in (0, 1)$ (since I can always find an $x : (\epsilon)^{1/n} < x < 1$) and therefore I have shown that $f_n (x) = x^n$ is not uniformly convergent in $(0, 1)$.

I am confused since I was asked to show that it is unformly convergent in $(0, 1)$. What is wrong?

2. Originally Posted by tholan
I am asked to show that $f_n (x) = x^n$ is uniformly convergent in $(0, 1)$ but not in $[0, 1]$.

I proceed by computing the pointwise convergence to $f(x) = 0$ for $x \in [0, 1)$ and $f(x) = 1$ for $x = 1$.

Then I compute for $x \in [0, 1)$:

$\vert f_n(x) - f(x) \vert < \epsilon \Leftrightarrow \vert x^n \vert < \epsilon \Leftrightarrow x < (\epsilon)^{1/n}$

This is not fullfilled for all $x \in (0, 1)$ (since I can always find an $x : (\epsilon)^{1/n} < x < 1$) and therefore I have shown that $f_n (x) = x^n$ is not uniformly convergent in $(0, 1)$.

I am confused since I was asked to show that it is unformly convergent in $(0, 1)$. What is wrong?
The question is wrong. The sequence of functions $f_n (x) = x^n$ is not uniformly convergent in $(0, 1)$, as the above argument correctly shows.

3. Thank you.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 30, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9720216989517212, "perplexity_flag": "head"}