# Suppose that $\{f_n\}$ is a sequence of real valued functions on $\mathbb{R}$.

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.

• 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. Dec 9 '20 at 21:33

You may want to try $$f_n(x)=f(x)=x$$.
What about $$f_n(x) = 0$$ for $$\vert x \vert \le n$$ and $$f_n(x) = n$$ otherwise?