Give an example of a decreasing function {$f_n$} of continuous functions on [0,1) that converges to a continuous limit function $f$, but the convergence is not uniform.
So far all my attempts either ends up in a discontinuous $f$ or the function is not decreasing.
Any help or insight is deeply appreciated.
You can take a sequence of function $(f_n)$ converging to $f:x\mapsto \frac{1}{x-1}$, defined in the following way : for $n \in \mathbb{N}$,
$$f_n\ \colon\ x\longmapsto\begin{cases} \frac{1}{x-1}&\text{if $0 \le x \le 1-\frac{1}{n+1}$}\\ -(n+1)&\text{otherwise} \end{cases}$$
All functions are defined on $[0,1)$. The $f_n$ and $f$ are continuous, and the decreasing sequence $(f_n)$ converges pointwise to $f$, but the convergence is not uniform because the $f_n$ are bounded and $f$ is not.