# Dini's theorem (specific case)

by Daniel98   Last Updated May 22, 2020 23:20 PM

Note: I asked this question before but it wasn't well written, So I deleted my previous question and re-wrote it.

According to Dini's theorem:

If $$X$$ is a compact topological space, and $$\{ f_n \}$$ is a monotonically increasing sequence (meaning $$f_n(x) \leq f_{n+1}(x)$$ for all $$n$$ and $$x$$) of continuous real-valued functions on $$X$$ which converges pointwise to a continuous function $$f$$, then the convergence is uniform.

The same conclusion holds if $$\{ f_n \}$$ is monotonically decreasing instead of increasing.

(Note: I have proven both cases)

But, what if for every $$n$$ $$\{f_n(x0)\}$$ is monotonic but for some values of $$n$$ it's monotonically decreasing and for other it's monotonically decreasing. for example; for all even values it is increasing and for non-even values it is decreasing.

How could I prove that Dini's theorem is effective in this case? In other words, how to prove that the convergence is uniform

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Let $$A=\{x: f_n(x) \leq f_{n+1}(x) \forall n\}$$ and $$B=\{x: f_n(x) \geq f_{n+1}(x) \forall n\}$$. Note that $$A$$ and $$B$$ are closed sets and hence they are also compact. Also $$A \cup B=X$$. $$f_n \to f$$ uniformly on each of these sets. Given $$\epsilon >0$$ there exist $$n_1, n_2$$ such that $$|f_n(x)-f(x)| <\epsilon$$ for all $$x \in A$$ for all $$n >n_1$$ and $$|f_n(x)-f(x)| <\epsilon$$ for all $$x \in B$$ for all $$n >n_1$$. Let $$n_0=\max {n_1,N_2\}. Then |f_n(x)-f(x)| <\epsilon for all x \in X for all n >n_0$$.