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.