Pick an $\epsilon < \dfrac{1}{2}$.
For example, say $\epsilon = \dfrac{1}{4}$.
Suppose, for the sake of showing a contradiction, that $\{f_n\}$ was uniformly convergent on $(0,\infty)$.
We then could find a natural number $N$ (having chosen $\epsilon$, this is now a FIXED natural number) such that:
$|f_n(x)| < \dfrac{1}{4}$ for all $n > N$ and all $x \in (0,\infty)$.
Let $x = \dfrac{1}{N+1}$.
We have:
$|f_{N+1}(N+1)| = \dfrac{1}{2} > \dfrac{1}{4}$, contradiction.
So $\{f_n\}$ must not be uniformly convergent, there is no such $N$.
What actually happens here, is we can find an $N$ that works on $(1/n,\infty)$, but no matter how big $N$ gets, there's still a little bit of $f$ for which the convergence is "too slow", even though that "little bit" winds up getting closer and closer to 0.
This example is closely related to the behavior of $f(x) = \dfrac{1}{x}$ near 0. Clearly, as we approach 0, $f$ approaches $\infty$, but no matter "how close" we get to 0, if we are not AT it, $f$ is still finite. We can bound $f$ on any interval:
$[\epsilon,1]$
but we cannot bound $f$ on $(0,1]$, even though it is defined everywhere on this interval.
This is essentially "built-in" to the real numbers, we can keep creeping closer to the cliff, and we're never actually "forced" to jump off. Even a tiny bit of "wiggle room" is enough to allow functions to do some very odd things.
