A note on equal convergence (Q1307387)

According to the reviewer and \textit{M. Laczkovich} [Acta Math. Hung. 55, No. 1-2, 165-178 (1990; Zbl 0718.54032)], a sequence of functions \(f_n\) into \(\mathbb{R}\) converges discretely to \(f:X\to \mathbb{R}\) iff, for each \(x\in X\), there is \(n_0 \in \mathbb{N}\) such that \(f_n(x)=f(x)\) whenever \(n>n_0\); it converges equally iff there exists a sequence \((\varepsilon_n)\) such that \(\varepsilon_n >0\), \(\varepsilon_n\to 0\) and, given \(x\in X\), there is \(n_0\in \mathbb{N}\) satisfying \(| f_n(x)-f(x)| <\varepsilon_n\) for \(n>n_0\). For a class \(\phi\) of real-valued functions on \(X\), let \(\phi^u\), \(\phi^d\), \(\phi^e\) denote the class of all uniform, discrete, equal limits of squences taken from \(\phi\). By answering a question posed in the above-mentioned paper, the author constructs a lattice \(\phi\) such that \(\phi^{ud}\subsetneq \phi^e\).