If $f_n(x) = \dfrac{x}{1+nx}$, show that ${f_n(x)}_{n=1, \cdots, \infty}$ converges uniformly to $0$ on $(0,1)$.
Here is what I have so far:
Let $\epsilon>0$ be given. Pick any $n \in \mathbb{N}$ and $\forall \epsilon >0$ let $N> 1/\epsilon$ (such an $N$ exists by the Archimedean property).
Then $|f_n(x) - f(x)| = |\dfrac{x}{1+nx} - 0| \ \ \forall n \geq N$.
$|\dfrac{x}{1+nx} | \leq 1/n ≤ \epsilon$ for all $x \in \mathbb{R}$.
Is this correct? Am I missing something?