Show the multiplication mapping of $S \times S \to S$ is not (jointly) continuous

Question

Let $S=R \cup {\alpha}$ be the one-point compactification of the usual space $R$ of real numbers. Define multiplication on $S$ by the rule $xy=x+y$ if $x$ and $y$ are in $R$, and $xy=\alpha$, otherwise.

Show the multiplication mapping of $S \times S \to S$ is not (jointly) continuous at the point $(\alpha, \alpha)$.

Thanks for your help!

Answer 1

HINT: For $n\in\Bbb N$ let $x_n=n$ and $y_n=-n$; then $\langle x_n:n\in\Bbb N\rangle$ and $\langle y_n:n\in\Bbb N\rangle$ both converge to $\alpha$. Does $\langle x_ny_n:n\in\Bbb N\rangle$ converge to $\alpha$?