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The problem is in Marsden - Elementary Classical Analysis 2nd ed. Ch.5.

My approach was

  1. Condition : $f_k, f(\in\mathcal C) : A\to N$, $f_k \to f \ \text{(pointwise)}$ and $A$ is compact.

  2. Since $A$ is compact, the range of $f_k, f$ are all compact, so bounded.

  3. 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?

Jinmu You
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    There are indeed counterexamples to this proposition. For example, $f_n$ could be the "spike" function of height $1$ and width $1/n$, centred on $1/n$. – TonyK Jun 05 '16 at 09:16
  • @TonyK Isn't that function not continuous? – Jinmu You Jun 05 '16 at 09:19
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    Perhaps I am using the wrong term. I mean the function whose graph is a straight line from $(0,0)$ to $(1/n,1)$ and a straight line from $(1/n,1)$ to $(2/n,0)$. (OK, that has width $2/n$, not $1/n$, but it will serve just the same.) – TonyK Jun 05 '16 at 09:42
  • @TonyK but this does not converge to a continuous function on a compact set. – dmh Aug 05 '22 at 18:15
  • @dmh: Yes it does. It converges to $0$ everywhere. – TonyK Aug 05 '22 at 18:50
  • I guess it's challenging for me to understand the limit here since the peak at 1 approaches 0, although the width converges to 0. – dmh Aug 05 '22 at 20:59
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    @dmh: just fix $x$, and see what happens to $f_n(x)$ as $n$ tends to infinity. The details are not difficult. – TonyK Aug 08 '22 at 00:10

1 Answers1

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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.