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.
Best Answer
The theorem as stated is true (see my comment though): If 1), 2), 3), and 4) hold, then $f_n$ converges uniformly to $f$.
The answer to your third question is "yes". This is the contrapositive of the theorem, which is logically equivalent to the theorem.
I think the answer to your second question is no (specifically, 4) does not necessarily have to hold).
Other observations:
Remark 1: The theorem is "tight" (?); that is, each of the hypotheses are needed to insure that the convergence is uniform.
To see this:
Take $K=[0,1]$. The functions $f_n(x)=x^n$ give a sequence satisfying 1), 3), and 4), but not 2).
$f_n$ converges to $f=0$ pointwise but does not converge uniformly to $f$ on $K$.
Take $K=[0,\infty)$ and $f_n(x)=\begin{cases} 0,\; & 0\leq x\leq n\\x-n , &n< x\leq n+1\\ 1 ,& x>n+1\end{cases}$. This sequence satisfies 1), 2) and 4) but not 3).
$f_n$ converges to $f=0$ pointwise but does not converge uniformly to $f$ on $K$.
Take $K=[0,1]$ and let $f_n(x)$ be the function whose graph consists of the straight line segment from $(0,0)$ to $({1\over2n},1)$, the straight line segment from $({1\over2n},1)$ to $({1\over n}, 0)$, and the straight line segment from $({1\over n},0)$ to $(1,0)$. Then $f_n$ satisfies 1), 2), and 3), but not 4).
$f_n$ converges to $f=0$ pointwise but does not converge uniformly to $f$ on $K$.
Take $K=[0,1]$ and $f_n(x)=\begin{cases}1, & 0\le x\le 1-{1\over n}\cr 0,&1-{1\over n}< x<1\\1,\;&x=1\end{cases} $. This sequence satisfies 2), 3) and 4) but not 1).
$f_n$ converges to $f=1$ pointwise but does not converge uniformly to $f$ on $K$.
Remark 2: If $(f_n)$ is uniformly convergent to $f$, you may not conclude that all four conditions hold. For example $f_n(x)=\begin{cases}1/n,& 0\leq x<1/2\\ -1/n& 1/2\leq x\leq1 \end{cases}$ converges uniformly to the zero function. But $(f_n)$ does not satisfy 1) or 4).
This shows that the converse of the theorem is false. It is not an "if and only if" theorem.
I will expand this post later (if that's allowed).