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Let $(a_n)$ be a decreasing sequence of continuous functions on the interval $[a,b]$ which converges pointwise to $0$. Show that it converges uniformly to $0$.

This is my attempt at the proof, could someone tell me if it is close to being right and if it's not (which I don't think that it is) could someone tell me how to approach proving this.

(I think this is a special case of Dini's theorem. However this is for a course where we have not done compactness or openness.)

Thanks for any help!

Proof.

From the definition of pointwise convergence, we have that:

$\forall \epsilon > 0 \quad \forall x \in [a,b] \quad \exists N: \quad \forall n \geq N: \quad |f_n(x)| < \epsilon $

Now let the max of $f_n(x)$ be at $x_0$. Then we have that:

$ \forall \epsilon > 0 \quad \exists N_0: \quad \forall n \geq N_0: \quad |f_n(x_0)| < \epsilon $

Now as $x_0$ is the max and $f_{N_0}>f_{N_0+1}> \cdots$, we have that:

$\forall \epsilon > 0 \quad \exists N_0 : \quad \forall n \geq N_0 \quad \forall x \in [a,b]: \quad |f_n(x)| < \epsilon $

So we have uniform convergence.

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    @hmmmm Following the idea of my first comment, but without using open covers, we will show that there is a $n_0$ such that $O_{n_0}=\left[a,b\right]$. If it's not the case, we can pick $x_n\in\left[a,b\right]\setminus O_n$ for all $n$, and by Bolzano-Weierstrass extract a converging subsequence $\{x_{n_k}\}$ which converges to a $x\in\left[a,b\right]$. What about $f_{n_k}(x)$?2011-11-01

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