Simultaneous Sequential Compactness

A set of sequences is said to converge simultaneously if there exists an infinite subset $H$ of the index set $\omega$ such that all sequences converge when restricted to $H$. We discuss simultaneous convergence of sequences in the same or in different sequentially compact spaces; we link the results for different spaces to ones for the same space; we show that simultaneous convergence happens for less than $\mathfrak{s}$ sequences in spaces with weight bounded by $\mathfrak{s}$ and for less than $\mathfrak{h}$ sequences in general; we show a slight generalisation of these results in the context of Hausdorff spaces; and finally we investigate their optimality.