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Let $\{f_n\}$ be a sequence of functions defined on $[0,1]$. Suppose that there exists a sequence of numbers $x_n$ belonging to $[0,1]$ such that $f_n(x_n)=1$.

Prove or Disprove the following statements.

  • a) True or false: There exists $\{f_n\}$ as above that converges to $0$ pointwise.
  • b) True or false: There exists $\{f_n\}$ as above that converges to $0$ uniformly on $[0,1]$.
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    Welcome to MSE! You might find a warmer response if you 1) reword the question to remove the imperative (and show that you aren't just a homework-copying robot), and 2) show some work2011-09-30
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    Please check the quantifiers. Should these be true for all sets $\{f_n\}$ or for some set $\{f_n\}$. My guess is for some set as $f_n=1$ for all $n$ and $x$ satisfies the condition and makes both false.2011-09-30
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    @Ross, the questions make sense if "as above" is taken to mean "such that the conditions in the first paragraph are satisfied".2011-09-30
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    @HenningMakholm: Arturo Magidin improved the wording after I put in this comment.2011-09-30

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