TITLE: Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?
QUESTION [9 upvotes]: The problem is in Marsden - Elementary Classical Analysis 2nd ed. Ch.5.
My approach was

Condition : $f_k, f(\in\mathcal C) : A\to N$, $f_k \to f \ \text{(pointwise)}$ and $A$ is compact.
Since $A$ is compact, the range of $f_k, f$ are all compact, so bounded.
Since $f_k, f\in\mathcal C_b$, if $\|f_k-f\|\to 0$ in $\mathcal C_b$ then $f_k\to f \ \text {(uniformly)}$.

when the norm of $\mathcal C_b$ is defined by
$\|f\|=\sup\{\|f(x)\||x\in A\}$.
I got stucked in 3, showing $\|f_k-f\|\to 0$. I don't even know the proposition is true. Can anyone give me some hint to solve this or a counterexample for this?

REPLY [10 votes]: As TonyK said, the fact that the limit is continuous does not imply the convergence is uniform. The sequence of functions $f_n(x)=\min(1,n|x-1/n|)$ converges to $1$ pointwise on the interval $[0,1]$, yet the convergence is not uniform. 
Dini's theorem states that monotone convergence of continuous functions on a compact set is uniform, provided the limit is continuous.