Uniform Convergent on $\mathbb{R}$

Question

Suppose $f_n:\mathbb{R}\rightarrow \mathbb{R}$ is a sequence of continuous functions, such that the sequence $f_n$ converges to $f$ uniformly on $[-N,N]$ for every fixed positive integer $N$. Which of the following must be true?

(b) $f_n$ converges to $f$ uniformly on $\mathbb{R}$.

(d) $f$ is continuous on $\mathbb{R}$.

These two statements, in my opinion, are both false. The other two statements (a) and (c) are just changing the uniform convergent set from $\mathbb{R}$ to $[-r,r]$ for any real numbers $r$.

(a) $f_n$ converges to $f$ uniformly on $[-r,r]$ for any fixed real number $r>0$

(c) $f$ is continuous on $[-r,r]$ for any fixed real number $r>0$.

These two I am sure that it is True since I can use Archimedean Principle to assert that I can find an integer $n$ such that $n>r$. However, this assertion should not be true in the question I asked in (b) and (d). I guess it must be wrong...but I don't know how to explain. Can anyone help me to explain what is the difference and how to explain this is a false statement?

Answer 1

(b) is false. (d) is true.

For (b) let $f_n(x)=x/n$ for $n\in \Bbb Z^+.$

For (d) $f_n$ converges uniformly to $f$ on $[-N,N]$ for all $N\in \Bbb Z^+$ and each $f_n$ is continuous on each $[-N,N],$ so $f$ is continuous on each $[-N,N] .$ So $f$ is continuous on $\Bbb R.$

Apply the following theorem with $X=[-N,N]$ and $Y=\Bbb R$ and $d(x,x')=|x-x'|$ and $e(y,y')=|y-y'|$:

Theorem: Let $(X,d), (Y,e)$ be metric spaces and let $(f_n:X\to Y)_{n\in \Bbb N}$ be a sequence of continuous functions converging uniformly (with respect to the metrics $d,e$) to $f:X\to Y.$ Then $f$ is continuous.