Are there any satisfying example of set $E(\varepsilon)$?

Question

I have a set $E = \left(0, 1\right]$ I would like to find an example of $E(\varepsilon)$ such as $\forall \varepsilon > 0$:

1. $E(\varepsilon) \subset E$
2. $|E(\varepsilon)| < \varepsilon$
3. sequence $\{f_{n}(x)\}$ where $f_{n}(x) = \frac{1}{nx}$ converges uniformly on $E\setminus E(\varepsilon)$

I found example $\left(0,\frac{1}{\varepsilon}\right)$ but its wrong for $\varepsilon = 0.0001$

Answer 1

Take $E(\varepsilon) =(0,\delta/2)$ where $\delta$ is the minimum of $\varepsilon $ and $1$. Then $f_n(x) \to 0$ uniformly on $E\setminus E(\delta)$.