Exercise 21.5

Proof. First, we have that the sequence $x_n\times y_n$ converges to $x\times y$ in the product space $ℝ\times ℝ$ by Exercise 19.6 since both $x_n\to x$ and $y_n\to y$ in $ℝ$. Now suppose that $f:ℝ\times ℝ\to ℝ$ is continuous. Then we have that

by Theorem 21.3. Now, we have that addition, subtraction, and multiplication are all continuous functions from $ℝ\times ℝ$ to $ℝ$ by Lemma 21.4. It then follows that for the continuous function $+:ℝ\times ℝ\to ℝ$ we have

It similarly follows that $x_n-y_n\to x-y$ and $x_n{y}_n\to \mathit{xy}$ as desired.

Regarding the quotient, we note that $(y_n)$ is a sequence in the subspace topology $\mathit{ℝnz}$ since each $y_n\ne 0$. We also note that $y\ne 0$ and hence also $y\in \mathit{ℝnz}$ so that the sequence $(y_n)$ converges to a point still within the space $\mathit{ℝnz}$. It then again follows that $x_n\times y_n\to x\times y$ in the product space $ℝ\times (\mathit{ℝnz})$ by Exercise 19.6. Since the quotient function is continuous from $ℝ\times (\mathit{ℝnz})$ to $ℝ$ by Lemma 21.4, it then follows as before that $x_n∕y_n\to x∕y$ by Theorem 21.3 just as we would like. □