uniform convergence over a countable union of sets

uniform convergence over a countable union of sets

I am working a problem on this book which asks to prove or disprove that
if $f_n \rightarrow f$ uniformly on $E_1, E_2, E_3, \dots,$ then $f_n
\rightarrow f$ uniformly on $\cup_{n=1}^\infty E_n$.
Two ideas come to mind: 1) If $E_n$ is decreasing, then surely the above
statement holds. 2) If a finite union was put in place of the countable
union, i.e. $\cup_{n=1}^k E_n$ then the proof for uniform convergence
would be straightforward seeing that we can take the maximum of a set with
a finite number of elements.
The second idea gives me the intuition that the statement above is not
necessarily true.
However I am finding it difficult to find counterexamples. I know that
$f_n(x)=x^n$ is not uniformly convergent on $[0,1]$ but that it is
uniformly convergent on $[0,\sigma]$ where $0\leq\sigma<1$. I cannot think
of sets whose countable union is $[0,1]$, unfortunately. Any hints on how
to go about constructing one, if possible?
Anyone has thoughts about how I should proceed?