Showing Uniform Convergence of $f_n(x) = \frac{nx}{1+nx}$ for $x \geq 0$.

Question

We've been going over Uniform convergence in my class, but I'm very uncertain as to how to effectively show the correct proof.

The $\lim_{n \rightarrow \infty}f(x) = 1$, understood using L'Hopital indeterminate form, which is not rigorously shown. (Help needed)

I'm having the most trouble showing that for $a > 0, \{f_n\}$ converges uniformly to $f$ on $[a, \infty)$.

Answer 1

Using l'Hôpital's rule for finding the limit function is a reasonable step to make. In order to investigate uniform convergence, we can use the "sup-criterion":

$$\sup_{x \in [a,\infty)} \lvert f_n(x)-f(x) \rvert=\sup_{x \in [a,\infty)} \left\lvert \frac{1}{1+nx} \right\rvert=\frac{1}{1+n a} $$ and the latter tends to $0$ as $n \to \infty$.