- #1
Chris L T521
Gold Member
MHB
- 915
- 0
Here's this week's problem!
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Problem: Let $\{f_n\}$ be an increasing sequence of continuous functions on $[a,b]$ which converges pointwise on $[a,b]$ to the continuous function $f$ on $[a,b]$. Show that the convergence is uniform on $[a,b]$.
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Hint: [sp]Let $\epsilon>0$. For each natural number $n$, define $E_n=\{x\in[a,b]\mid f(x)-f_n(x)<\epsilon\}$. Show that $\{E_n\}$ is an open cover of $[a,b]$ and use the Heine-Borel theorem.[/sp]
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
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Problem: Let $\{f_n\}$ be an increasing sequence of continuous functions on $[a,b]$ which converges pointwise on $[a,b]$ to the continuous function $f$ on $[a,b]$. Show that the convergence is uniform on $[a,b]$.
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Hint: [sp]Let $\epsilon>0$. For each natural number $n$, define $E_n=\{x\in[a,b]\mid f(x)-f_n(x)<\epsilon\}$. Show that $\{E_n\}$ is an open cover of $[a,b]$ and use the Heine-Borel theorem.[/sp]
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!