Uniform convergence on closed interval.

Question

Suppose that $$\{f_{n}\} $$ converges uniformly on $[0,a]$ for all positive number $a$. Then, the function sequence converges uniformly on $$[0,\infty).$$

Is the above statement true? I have no idea..

Answer 1

No: let $f_n(x):=e^{-(x/n)^2}$. Then $f_n\to 1$ uniformly on $[0,a]$ for all $a$, but $f_n \not \to 1$ uniformly on $[0,+\infty)$.

Answer 2

No. If $f_n(x)=\frac xn$, then $(f_n)_{n\in\mathbb N}$ converges uniformly to $0$ on every interval $[0,a]$ ($a>0$), but not on $[0,+\infty)$.

Answer 3

No. Let $f_n(x)=\max\{0,1-|x-n|\}$. Then $f_n\to 0$ uniformly on $[0,a]$ for all $a$ (in fact, almost all $f_n$ are $\equiv 0$ on $[0,a]$), but not on $[0,\infty)$.