Bounded limit inferior and limit superior of functions on dense sets implies divergence on uncountable dense set.

Question

Reading an article by Akcoglu et al. "The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters", which can be found here:

http://journals.cambridge.org/abstract_S0143385700008798,

I came across a claim of the form: Let $(f_{n})$ be a sequence of continuous functions on $[0,1)$ with the Lebesgue measure, let $A,B\subset [0,1)$ be dense sets and $a,b \in [0,1)$ $a>b$ such that for all $x \in A$ $\liminf_{n\rightarrow\infty}f_{n}(x) \geq a$ and for all $x \in B$ $\limsup_{n\rightarrow \infty} f_{n}(x) \leq b$. Then on an uncountable dense set of $x$'s the limit $\lim_{n\rightarrow\infty}f_{n}(x)$ does not exist.

I am having trouble proving this result (or even seeing why it should be true). According to the paper it follows from the Baire category theorem, but I am not too sure how that helps. Also we do not have that our sequence of functions are uniformly continuous which makes it more difficult.

Any ideas, help or references would be much appreciated.

Answer 1

So I made some progress on this problem and managed to find a dense set, but I am not sure how to prove the uncountable part. This fact was however sufficient for me. Proof as follows:

Without loss of generality we can assume that bounds on the limsup and liming are strict. Define $A_{n} := \{x : f_{n}(x) > a \}$ and $B_{m} := \{x : f_{m}(x) < b \}$. As the $f_{n}$ are continuous, $A_{n}$ and $B_{m}$ are open sets. $\bigcup_{n=N}^{\infty}A_{n}\supset A$ and $\bigcup_{m=M}^{\infty}B_{n}\supset B$, so $\bigcup_{n=N}^{\infty}A_{n}$ and $\bigcup_{m=M}^{\infty}B_{n}$ are open dense sets. From the Baire category theorem the countable intersection \begin{align*} \bigcap_{N=1}^{\infty}\bigcap_{M=1}^{\infty}\left(\bigcup_{n=N}^{\infty}A_{n}\right)\cap\left(\bigcup_{m=M}^{\infty}B_{m}\right) & = \bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty}\bigcap_{M=1}^{\infty}\bigcup_{m=M}^{\infty}\left(A_{n}\cap B_{m}\right)\\ &= \left\{x : \limsup_{N\rightarrow \infty}f_{N}(x) > a \text{ and } \liminf_{N\rightarrow\infty}g_{N}(x)< b\right\} \end{align*} is dense.