This result follows from much stronger results from general topology. These results can be found in [1].


> $\mathbf{Theorem}$ Each non-discrete closed subset of $\beta X\setminus\upsilon X$ 
> contains a copy of $\beta\mathbb{N}$ (and in
> particular, its cardinality is at least $2^{2^{\aleph_{0}}}$). 

$\phantom{Here is a secret message.}$
> $\mathbf{Corollary}$ If $X$ is locally compact and realcompact, then every infinite
> closed set in $\beta X\setminus X$ contains a copy of $\beta\mathbb{N}$ (thus, its cardinality is at least $2^{2^{\aleph_{0}}}$).

As a result of the above corollary, since every discrete space of non-measurable cardinality is realcompact, if $A$ is a discrete space of non-measurable cardinality, then every closed set in $\beta A\setminus A$ contains a copy of $\beta\mathbb{N}$.

Now assume that $A$ is a set of non-measurable cardinality and $(x_{n})_{n}$ is a convergent sequence in $\beta A$ that converges to some point $x\in\beta A$. Take a subsequence $(y_{n})_{n}$ such that $\{y_{n}|n\in\mathbb{N}\}\subseteq A$ or $\{y_{n}|n\in\mathbb{N}\}\cap A=\emptyset$. If $\{y_{n}|n\in\mathbb{N}\}\subseteq A$, and $\{y_{n}|n\in\mathbb{N}\}$ takes infinitely many values, then take a subsequence $(z_{n})_{n}$ where each $z_{n}$ is distinct. Let $f:A\rightarrow[0,1]$ be a function where $z_{n}=0$ whenever $n$ is even and $z_{n}=1$ whenever $n$ is odd. Then $f$ extends to a continuous $\overline{f}:A\rightarrow[0,1]$. In particular, $\overline{f}(z_{n})\rightarrow f(x)$. This is a contradiction since $\overline{f}(z_{n})$ oscillates between $0$ and $1$ endlessly. Therefore $\{y_{n}|n\in\mathbb{N}\}$ can only take finitely many values, so $(y_{n})_{n}$ is eventually constant.

On the other hand, if $(y_{n})_{n}\subseteq\beta A\setminus A$, then since $\beta A\setminus A$ is closed, we have $x\in\beta A\setminus A$ as well. Therefore $\{y_{n}|n\in\mathbb{N}\}\cup\{x\}$ is a closed subset of $A$ of cardinality less than $2^{2^{\aleph_{0}}}$. Therefore, by the above corollary, the set $\{y_{n}|n\in\mathbb{N}\}\cup\{x\}$ is finite, so the sequence $(y_{n})_{n}$ must be eventually constant.



[1] Gillman, Leonard, and Meyer Jerison. Rings of Continuous Functions,. Princeton, NJ: Van Nostrand, 1960.