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Suppose that $\{f_n\}$ is a sequence of real valued functions on $\mathbb{R}$. Suppose it converges to a continuous function $f$ uniformly on each closed and bounded subset of $\mathbb{R}$. Which of the following statements are true?

  1. The sequence $\{f_n\}$ converges to $f$ uniformly on $\mathbb{R}$.

  2. The sequence $\{f_n\}$ converges to $f$ pointwise on $\mathbb{R}$.

  3. For all sufficiently large $n$, the function $f_n$ is bounded.

  4. For all sufficiently large $n$, the function $f_n$ is continuous.

I have discarded options 1, 4 by choosing $f_n(x)=0$ for all $|x|\leq n$ and 1 otherwise. Further option 2 is true, by the given condition. please give a hint for option number 3.

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  • $\begingroup$ What if you take $f_n=f$ for all $n$, then obviously $\{f_n\}$ converges uniformly to $f$ on each closed and bounded subset of $\mathbb{R}$. It suffices to chose a $f$ that is not bounded. $\endgroup$
    – Tuvasbien
    Dec 9 '20 at 21:33
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You may want to try $f_n(x)=f(x)=x$.

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What about $f_n(x) = 0$ for $\vert x \vert \le n$ and $f_n(x) = n$ otherwise?

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