So I have this question in my math homework, and I really have no idea how to prove it, could someone please help me?
Let $A\subset \mathbb R$ and $B\subset \mathbb R$ be two compact sets such that $\alpha = \inf B$ satisfies $\alpha > 0$.
Let $A/B = \{x/y \mid x\in A \text{ and } y \in B\}$. Prove that $A/B$ is compact.
My definition of compactness is the following:
Let $X$ be a metric space. A set $E \subset X$ is compact if and only if for each open cover $\{O_\alpha\}_{\alpha\in I}$ of $E$ indexed by some set $I$, there is an $n\in \mathbb{N}^*$ and indices $\alpha_i \in I$ such that $E\subset\bigcup_{i=1}^n O_{\alpha_i}$.
Sorry for the bad writing, but I don't know how to use the math notations.