Def (Sequential Accumulation Point): Given a topological space $(X,\tau)$ and a subset $S\subseteq X$, we say that $a\in X$ is a sequential accumulation point if there exists $(s_n)_{n\geq1}\subseteq S\setminus\{a\}$ that converges to $a$.
The name comes from the following analogy. We say that $a\in X$ is an accumulation point of $S\subseteq X$ whenever $a\in\text{cl}_X(S\setminus\{a\})$. So, if we consider the sequential closure of a set (i.e. the set of points in $X$ that can be expressed as a limit of a sequence of the said set), we have that $a\in X$ is a sequential accumulation point of $S$ whenever $a\in\text{scl}_X(S\setminus\{a\})$.
Now, we know that in compact Hausdorff spaces every infinite subset must have an accumulation point. My question then is
Q: Must every infinite subset $S\subseteq X$ of a compact Hausdorff space have a sequential accumulation point?
Note that, if true, this fact would be stronger as any sequential accumulation point is an accumulation point since $\text{scl}_X(S\setminus\{a\})\subseteq\text{cl}_X(S\setminus\{a\})$. Also, the answer to this question is affirmative if $X$ is first-countable as first-countable implies Fréchet-Urysohn. But not every compact Hausdorff space is first-countable. If there's a counterexample, could $S\subseteq X$ be $G_\delta$ with no sequential accumulation point if $X$ is compact and Hausdorff?
For one definition of a Stone-Čech compactification and some of its properties see Eduard Čech 'On Bicompact Spaces' section II.
For a proof that projective objects in the category of compact Hausdorff spaces are extremally disconnected see Andrew Gleason 'Projective Topological Spaces' Theorem 1.2.
For a proof that sequences in extremally disconnected spaces are eventually constant see same paper by Gleason, Theorem 1.3.
Proving that $\beta\Bbb{N}$ is projective in the category of compact Hausdorff spaces is easy from the universal property of a Stone-Čech compactification, but if you can't figure it out, it's in John Rainwater 'A Note on Projective Resolutions' Lemma 3.
A projective object in the category of compact Hausdorff spaces is one that has a similar universal property for projective objects in algebra - i.e. projective module over a ring. An epimorphism in topological spaces is a surjective continuous map.