Let $y\in Y\subset \mathbb{R}^n$, $t\in T\subset \mathbb{R}$. Let $T$ and $Y$ be compact sets. Define the set $S=\{ty\in \mathbb{R}^n\mid t\in T, y\in Y\}$. Is $S$ compact?
${\bf Question:}$ Is $S$ compact? I believe so and the following is my proof. Is my proof correct?
${\bf Proof:}$ Define a function $f:T\times Y\mapsto \mathbb{R}^n$, such that $f(t,y)=ty$. First, I will show that it is continuous.
$|t_ny_n-ty|=|t_ny_n+ty_n-ty_n-ty|\\ \hspace{0.8in}=|(t_n-t)y_n+t(y_n-y)|\\ \hspace{0.8in}\leq |t_n-t|\;\lVert y_n\rVert +|t|\;\lVert y_n-y\rVert\to 0$.\
If $T$ and $Y$ are compact, $T\times Y$ is compact. Thus, $S$ is a continuous image of a compact set. Therefore, it is compact.