How to visualize Dini theorem regarding sequence of functions?

Question

I recently encountered Dini theorem while studying sequence of functions.But the proof is seeming tasteless.I understood the proof that we find in books but I could not interpret it graphically,I want to understand what it means graphically.Can anyone suggest me some visual interpretation behind Dini theorem that would give me a better insight of that theorem or so that I can develop independently my own thought process to prove the theorem.

Answer 1

"Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform."

My (poor) intuition for this result is as follows:

Uniform convergence means that the rate of convergence can be lower bounded. Now since the sequence of functions is pointwise increasing (or decreasing) they are all squeezed in a smaller and smaller gap towards the continuous limit. Furthermore, continuous functions on compact spaces have nice properties (uniform continuity and so forth).

This might not help much yet -- but in analysis, when dealing with problems that ask to upper bound some quantity in order to make it smaller than $\epsilon$ (using the triangle inequality), I personally find it more helpful to summarise the data of the problem to see how you can use it with the triangle inequality to upper bound your quantity of interest, rather than trying to visualise it. Finding the solution often helps in visualising the problem a posteriori.

Here you want to show that the sup norm on your compact space between your function sequence $f_n$ and its limit $f$ decreases as $n\to \infty$. Can you combine the data of your problem to upper bound $\sup_{x\in K}d(f_n(x),f(x))$, using the triangle inequality multiple times? HINT:at some point you might need to subdivide your compact space into very small compact subspaces.

Answer 2

Lat $(f_n)_{n\in\mathbb N}$ be a sequence of continuous functions from $[a,b]$ into $\mathbb R$ such that each sequence $\bigl(f_n(x)\bigr)_{n\in\mathbb N}$ ($x\in[a,b]$) is decreasing and converges to $f(x)$, for some continuous function $f\colon[a,b]\longrightarrow\mathbb R$. Now, let $g_n(x)=f_n(x)-f(x)$. Then each $g_n$ is continuous and the sequence $(g_n)_{n\in\mathbb N}$ converges pointwise to the null function. Furthermore, each sequence $\bigl(g_n(x)\bigr)_{n\in\mathbb N}$ ($x\in[a,b]$) is decreasing.

For each $n\in\mathbb N$, let$$G_n=\left\{(x,y)\in[a,b]\times\mathbb R\,\middle|\,0\leqslant y\leqslant g_n(x)\right\}.$$Then $(G_n)_{n\in\mathbb N}$ is a decreasing sequence of bounded subsets of $[a,b]\times\mathbb R$; being a decreasing sequence of sets is a consequence of the fact that each sequence $\bigl(g_n(x)\bigr)_{n\in\mathbb N}$ is decreasing. Furthermore,$$\bigcap_{n\in\mathbb N}G_n=[a,b]\times\{0\}.$$It is not hard to deduce from this that, given $\varepsilon>0$, if $n\gg1$, then $G_n\subset[0,b]\times[0,\varepsilon)$.