Suppose $f_n$ is a sequence of functions defined a set $K$ with pointwise limit function $f$.
I am confused about the following.
If the following conditions are satisfied:
- $f_n$ is continuous on $K$ for all $n$.
- The pointwise limit $f$ is continuous on $K$.
- $K$ is a compact interval (i.e., a closed and bounded interval in $\mathbb{R}$).
- The convergence of $f_n$ to $f$ is increasing or decreasing.
Then does this imply that $f_n$ is uniformly convergent to $f$?
Now a different problem: If one of these conditions is not satisfied, does this imply that $f_n$ is not uniformly convergent to $f$?
If $f_n$ is not uniformly convergent to f, does this mean that one these four conditions doesn't hold?
In general: what is the logical relationship between uniform convergence and these four conditions?
Please, I need answers to all the above questions because this theorem always confuses me when I solve the problems and I don't know how to use it properly. Thanks for your help in advance.