I am new to MSE, so apologies for any inconveniences regarding my question. I am studying for the GRE's Math Subject and decided to go back and study certain important concepts from the Analysis series. I am reading Elementary Analysis by Ross, particularly section 24, Uniform Convergence.
We are shown the sequence of real valued functions $$f_n(x) = nx^n, \quad x \in [0,1),$$ and told that it does not converge uniformly on said interval. However, $\ f_n \rightarrow f$ point-wise on $[0,1)$ where $\ f(x) = 0, \ \forall x \in [0,1).$
This all makes sense and I am fine with it. However, it is later remarked that a sequence $\ (f_n)$ of functions on a set $\ S \in \mathbb{R}$ converges uniformly if and only if $$lim[sup\{|f(x)-f_n(x)|:x \in S\}] = 0.$$
This is what confuses me, because I am pretty sure (not $100 \%$ sure), that since $limsup = L = liminf$ for any convergent sequence (where $L$ is the limit), then for the example $f_n(x) = nx^n$, shouldn't this sequence of functions technically be uniformly convergent?
I'm most definitely missing something, though hopefully not anything glaringly obvious, and would appreciate some feedback. Thanks!