Exercise 1.5.13

Question

Let $E$ and $F$ be two compact subsets of $\mathbb{R}$ (with the standard metric $d(x, y)=|x-y|$ ). Show that the Cartesian product $E 	imes F$ is a compact subset of $\mathbb{R}^{2}$ (with the Euclidean metric $d_{l^{2}}$ ).

Prakash Anand · Accepted Answer

Consider the sequence $\left\{\left(x_{n}, y_{n}ight)ight\} \in E 	imes F$, our goal is to find a convergent subsequence in $E 	imes F$.\par

Let $\left(x_{n}ight) \subset E$. Since $E$ is compact, we know there is a convergent subsequence in $E$. That is, $x_{n_{j}} ightarrow x \in E$.

Let $\left(y_{n_{j}}ight) \subset F$. Since $F$ is compact, we know there is a convergent subsequence in $F$. That is, $y_{n_{j_{k}}} ightarrow y \in F$.\par

Now we also have that $x_{n_{j_{k}}} ightarrow x \in E$. (Lemma 1.4.3)\par

Now we can use Proposition 1.1.18 to get that $\left(x_{n_{j_{k}}}, y_{n_{j_{k}}}ight) ightarrow(x, y) \in E 	imes F$

Exercise 1.5.13

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Exercise 1.5.13

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