There are $ A $ and $ B $ subsets of $ \mathbb{R} $, defined $ AB = \{ ab: a \in A, b \in B\} $. Now suppose that $ A $ and $ B $ are compact sets, then prove that $ AB $ is a compact set.
I took a sequence $ z_ {n} \in AB $ must then exist $ x_ {n} \in A $ and $ y_ {n} \in B $ such that $ z_{n} = x_{n} y_{n}$, then use their successions are subsequences, however,i do not know if $ z_ {n} $ can be written that way.
That is my question. any suggestion is appreciated
You can write each $z_n$ in that way. Since $A$ is compact, there exists a convergent subsequece $(x_{n_j})$. Now, the subsequence $(y_{n_j})$, also admits a convergent subsequence $y_{n_{j_k}}$. So the sequence $(z_n)$, admits a convergente subsequence $z_{n_{j_k}}$, and $AB$ is compact.