# Clue or hint on how to proceed with this Real Analysis exercise on uniform convergence

I am self-learning Real-Analysis. I'd like a clue or hint on how to proceed with this exercise problem from Stephen Abbott's Understanding Analysis without revealing the entire proof.

[Abbott, 6.2.10] This exercise and the next explore partial converses of the Continuous Limit Theorem (Theorem 6.2.6). Assume that $$f_n \to f$$ pointwise on $$[a,b]$$ and the limit function $$f$$ is continuous on $$[a,b]$$. If each $$f_n$$ is increasing (but not necessarily continuous), show that $$f_n \to f$$ uniformly.

Proof Attempt.

I tried a direct proof, but I am not exactly sure how to use the fact that each $$f_n$$ is increasing.

I tried a proof with contradiction.

We are given that, $$f_n \to f$$ pointwise on $$[a,b]$$ and the limit function $$f$$ is continuous on $$[a,b]$$. Further each $$f_n$$ is increasing. We proceed by contradiction.

Assume that $$f_n$$ does not converge uniformly to $$f$$. Thus,

\begin{align*} (\exists \epsilon_0 > 0), (\forall N \in \mathbf{N}) : (\exists n_0 \geq N),(\exists x_0 \in [a,b]) : |f_n(x_0) - f(x_0)| \geq \epsilon_0 \end{align*}

Since, $$f$$ is continuous on the compact set $$[a,b]$$, it is uniformly continuous on $$[a,b]$$. Now, I am not sure, how to proceed.

Any hints would be super-helpful guys.

• Yes this one is rather known as “false Dini theorem”. It is in fact a result due to Pólya. See the French Wikipedia page about this. Jan 14, 2022 at 16:36
• Asked and answered many times on this site — for example, here.
– RRL
Jan 14, 2022 at 16:57
• But the OP sates clearly that he/she want hints, not the full solution so it's no surprise that he/she is not googling it Jan 15, 2022 at 0:07

$$1).\ f$$ is increasing.
$$2).\$$ There is a partition $$\{a,x_1,\cdots x_{n-2},b\}$$ such that $$f(x_i)-f(x_{i-1})<\epsilon;\ 0\le i\le n.$$
$$3).\$$ if $$x\in [a,b]$$ then $$x_i\le x\le x_{i-1}$$ for some $$0\le i\le n$$ and $$f_n(x_{i-1})-f(x_{i-1})-\epsilon\le f_n(x)-f(x)\le f_n(x_i)-f(x_i)+\epsilon.$$
$$4).\$$ There is an integer $$N$$ such that $$|f_n(x_i)-f(x_i)|<\epsilon$$ for each $$0\le i\le n$$ and $$n>N.$$