Can anyone find a sequence of arbitrary functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, such that for each $n$, there is a sequence of arbitrary functions $f^{k}_{n}: \mathbb{R} \to \mathbb{R}$ that converges pointwise to $f_n$ as $k$ goes to $\infty$, but such that no subsequence of the $f^{k}_{n}$'s converges to $f$? Alternatively, can it be shown that such an example is impossible?
(These functions cannot be measurable, otherwise there is a simple argument that shows that this cannot work.)
The question arises as follows: suppose $A$ is set of functions $\mathbb{R} \to \mathbb{R}$, $B$ is a subset of $A$ such that every function in $A$ is the pointwise limit of functions from $B$, and $C$ is a subset of $B$ such that every function in $B$ is the pointwise limit of functions from $C$. Is every function in $A$ a pointwise limit of functions from $C$? The obvious approach is to use a `diagonal subsequence' argument.
Note that this is not the same as showing that a dense subset of a dense subset is dense in the whole space in the topological sense, since the topology of pointwise convergence is not first-countable, and so the sequential closure differs from the `setwise' closure.
It seems the following.
The space $X_0$ of arbitrary functions $f : \mathbb{R} \to \mathbb{R}$ endowed with the topology of pointwise convergence is just the Tychonov product $\mathbb{R}^\frak c$. Thus in order to construct a required counterexample $X\subset X_0$ it suffices to construct a Tychonov (that is completely regular) counterexample $X$ of weight not greater than $\frak c$. Then there is a homeomorphic embedding of the space $X$ into the product $\mathbb{R}^\frak c$. And such an example of the space $X$ is well-known.
(cited from “General Topology” by Ryszard Engelking. )