Difference in interval notation What is the difference between these two exercise questions (regarding the intervals)?

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*Show that for each $\epsilon > 0$, $f_n$ is uniformly convergent on $[\epsilon, \infty)$


*Show that $f_n$ is not uniformly convergent on $(0, \infty)$
I mean in the first interval we don't have the 0 because $\epsilon > 0$ and in the second we don't have the 0 because it is an open interval.
So what is the difference?

Consider the sequence functions $\{f_n\}$ defined on $\mathbb{R^+}$ by
$$f_n(x) = \dfrac{1}{nx}$$
Now for a fixed $\epsilon > 0$, $\{f_n\}$ converges uniformly on the interval $[\epsilon, \infty)$ because $|f_n(x) - f_m(x)| =  \frac{1}x|\frac{1}n - \frac{1}{m}| \leq $ $\frac{1}\epsilon |\frac{1}n - \frac{1}{m}|$ gets arbitrarily small for sufficiently large $n,m$ (i.e., $\{f_n\}$ is a Cauchy sequence). But if you consider the interval $(0, \infty)$, where $1/x$ gets arbitrarily large, the choice of $n$ and $m$ depends on the value of $x$.
